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Probability Theory - Science topic

The analysis of random phenomena and variable, stochastic processes and modeling.
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Probability theory in number theory and complex analysis I have obtained some scientific conclusions and results. These are the finite or infinite condition of the existence of the uyechim for Gilbert's 10th problem, flaws in Bernstein's views on disinhibition (tetrahedral), and alternative annotations to it, determining the defects of the Byuffon problem, and alternative solutions to it, the specifying function detection algorithm for Fibonacci-type sequences. I would like to study at the doctoral (PHD) now. Can you give me some advice?
I do not know English very well, sorry for the shortcomings
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Thank you Auroro, for attention!
I love pure (classical ) mathematics. Programming is not interesting to me. Today, for some reason, mathematics has become an abstract science, the essence has been forgotten. I want to waste my life to classical mathematics, I do not like formality. Unfortunately today, a flexible (non-haracteristic) person is preferred by most people. What I want to say in a general sense is that I only need a chance to study.
Focusing on solving great mathematical problems, their accent is completely different from the conclusions of scientists who tried at first. unusual look and a new finished solution. Private solutions, which are seen as commonplace in some cases, are only distracting. I believe that there is an infinite way to solve any problem. The one who tries to commit will definitely do it. We can't teach hechkim how to learn mathematics, what scientific results to achieve, this is from the God-given talent.
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What’s the most common programming paradigm of no-code platforms? Why?
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The most common programming paradigm of no-code platforms is declarative programming.
Why Declarative Programming?
  • Focus on "What" not "How": No-code platforms allow users to describe what they want to achieve (e.g., "create a form," "generate a report") rather than how to do it. This makes it accessible to users who may not have traditional programming skills.
  • Visual Interfaces: These platforms often provide drag-and-drop interfaces, workflows, and rule-based systems, which align well with declarative principles where users specify the desired outcomes rather than writing detailed procedural code.
  • Ease of Use: Declarative paradigms reduce the complexity involved in traditional coding. Users can build applications, workflows, and automations by configuring pre-built components or templates, which is ideal for business users or non-developers.
  • Abstraction: Declarative programming abstracts the underlying logic and implementation details. This allows the platform to handle complex tasks behind the scenes, which is crucial for empowering users without deep technical knowledge.
These factors make declarative programming the ideal paradigm for no-code platforms, supporting their goal of democratizing software development and making it more accessible to a broader audience
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No, tail risks do not prompt cryptozoology.
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In my opinion this approach is true. Because short size texts are more attractive for people, due to the fact that they can consider that they will read it in a few minutes and understand easily. Actually, to produce short and understandable texts in academy is not that simple ( because we want to write everything clearly ) , readers and researchers need this type of readings.
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Yes, so they should follow DEI.
1)
Preprint Nuance
2)
Preprint Nuance 2
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Reconnecting by Reading: A Selection of Books About Diversity, Equity, Inclusion and Accessibility Part 2
"Today we continue our recommendations for reading about diversity, equity, inclusion, and accessibility... As we continue to reconnect, it is our hope that you have books you may wish to recommend for continued growth and learning. Please post them in the comments and we thank you in advance for sharing them. In 2020, SSP “Reaffirmed our Commitment to Diversity, Equity and Inclusion” and will continue on that path via the work of the SSP DEIA Committee and our active and engaged membership..."
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The vastness or sustainability of a belief system doesn't necessarily indicate its plausibility. Plausibility depends on evidence, logical coherence, and consistency with observable phenomena, rather than the popularity or longevity of a belief. While a belief system's widespread acceptance or enduring nature may influence perceptions, plausibility is determined by its ability to withstand critical scrutiny and align with empirical reality.
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House-selling is one of the typical tasks of the Optimal Stopping problems. Offers come in daily for an asset, such as a house, that you wish to sell. Let Xi denote the amount of the offer received on day i. X1,X2,... are independent random variables, according uniform distribution on the interval (0...1). Each offer costs an amount C>0 to observe. When you receive an offer Xi, you must decide whether accept it or to wait for a better offer. The reward sequence depends on whether or not recall of past observations is allowed. If you may not recall past offers, then Di(X1,...,Xi)=Xi – i*C. If you are allowed to recall past offers, then Di(X1,...,Xi)=max(X1,...,Xi) – i*C. These tasks may be extended to infinite horizon (i is unlimited). So, there 4 different task statements :
  • without recall, infinite horizon
  • without recall, finite horizon
  • with recall, infinite horizon
  • with recall, finite horizon
First three tasks are quite simple, but I was unable to prove solution of the last task (in strict form, although I found a solution). If anyone knows her solution, please write it or send an article (link to the article) where it is written. Thank you in advance.
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Hello, I would suggest to first understand the problem statement. The bid prices cannot be uniformly distributed on the interval from 0 to 1. The bid prices have a typical kind of distribution (log-normal). You can study it at once. This will affect the solutions you get. I see no need to formalise the price range to the interval 0,1. That's really my personal non-binding opinion
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good question. But I have no comments yet.
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How can probability theory and statistical modeling contribute to our understanding of phonological variation and probabilistic phonological processes?
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Check out these few articles.
file:///C:/Users/user/Downloads/aldereteEtAl_21_Probabili.pdf
The hand of statistics and probability is open to express and present any type of model and probability theory.
What matters is your ideas and data.
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I am looking for a book, that would explain in a pretty simple language the different statistical methods and their practical application in the research, the social research in particular. The book is needed for a person who does not have a prior knowledge in statistics or probability theory.
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For those who are not adequately familiar with statistics, it is better to avoid "facilitated books".
In these cases I recommend various tutorials, which are easily available on the net, specific to each individual topic.
There are several on youtube.
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Suppose A is a set measurable in the Caratheodory sense such for n in the integers, A is a subset of Rn, and function f:A->R
After reading the preliminary definitions in section 1.2 of the attachment where, e.g., a pre-structure is a sequence of sets whose union equals A and each term of the sequence has a positive uniform probability measure; how do we answer the following question in section 2?
Does there exist a unique extension (or method constructively defining a unique extension) of the expected value of f when the value’s finite, using the uniform probability measure on sets measurable in the Caratheodory sense, such we replace f with infinite or undefined expected values with f defined on a chosen pre-structure depending on A where:
  1. The expected value of f on each term of the pre-structure is finite
  2. The pre-structure converges uniformly to A
  3. The pre-structure converges uniformly to A at a linear or superlinear rate to that of other non-equivalent pre-structures of A which satisfies 1. and 2.
  4. The generalized expected value of f on the pre-structure (an extension of def. 3 to answer the full question) satisfies 1., 2., and 3. and is unique & finite.
  5. A choice function is defined that chooses a pre-structure from A that satisfies 1., 2., 3., and 4. for the largest possible subset of RA.
  6. If there is more than one choice function that satisfies 1., 2., 3., 4. and 5., we choose the choice function with the "simplest form", meaning for a general pre-structure of A (see def. 2), when each choice function is fully expanded, we take the choice function with the fewest variables/numbers (excluding those with quantifiers).
How do we answer this question?
(See sections 3.1 & 3.3 in the attachment for an idea of what an answer would look like)
Edit: Made changes to section 3.5 (b) since it was nearly impossible to read. Hopefully, the new version is much easier to process.
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Einstein was also determined to answer questions he found worth pursuing, so, continue studying , reading, writing, and , sharpen your mind by studying published refereed papers as well, and then, after maybe quite some time, you really know, whether what you are doing is really worthwhile , and then, knock on the door of a professor.....and YOU should tackle your questions......and, also very important, when you write a research paper, introduce your problem carefully, in such a way, that your paper triggers the minds of those, who are reading your paper, and, do not write , in the beginning , in a terse style.
Good luck, and take your time, as did Einstein and Gödel or Hilbert!
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The central limit theorem for triangular arrays is well-known (see Durrett, Probability: Theory and examples, 4th ed., Theorem 3.4.5). Is there a local form of this theorem for integer-valued random variables? A local central limit theorem for sums of i.i.d. integer-valued random variables is provided by Gnedenko and Kolmogorov (see Gnedenko and Kolmogorov, Limit distributions for sums of independent variables, Chapter 49). Has the latter been generalized to triangular arrays? I am pretty sure that this has been done.
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I got the answer! The problem is addressed in
- McDonald, D.R. (1980), On local limit theorem for integer-valued random variables. Theory Probab Appl., 3, 613-619
- Mukhin, A.B. (1984), Local limit theorems for distributions of sums of independent random vectors. Theory Probab Appl. 29, 369-375
- Mukhin, A.B. (1991), Local limit theorems for lattice random variables. Theory Prob. Appl. 35, 698-713
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The concept of propensity was first introduced by Popper (1957). It is considered as an alternative interpretation of the mathematical concept of probability. Bunge (1981) examined the mathematical concept of probability and its personalist, frequentist, and propensity interpretations. He stated,
“The personalist concept is invalid because the probability function makes no room for any persons; and the frequency interpretation in mathematically incorrect because the axioms that define the probability measure do not contain the (semiempirical) notion of frequency. On the other hand the propensity interpretation of probability is found to be mathematically unobjectionable and the one actually employed in science and technology and compatible with both a possibilist ontology and a realist epistemology.”
However, it seems that the concept of propensity has been forgotten for many years. I've recently read several papers on propensity and found that it may be a useful concept for measurement uncertainty analysis, rather than the concept of "degree of belief" of Bayesian approaches to measurement uncertainty analysis.
BungeM 1981 Four concepts of probability Applied Mathematical Modelling 5(5) 306-312
Popper K R 1957 The propensity interpretation of the calculus of probability and the quantum theory, In S. KSrner (ed.), Observation and interpretation. Butterworths, London.
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Gang (John) Xie Thanks for your input. I think subjective probability or degree of belief may be suitable for some fields such as social science, it is not suitable for other fields such as measurement science. For an example, the normal distribution is also called the law of error. That is, the normal distribution is a physical law that describe the intrinsic property of a measurement system; it exists objectively, independent of person's mind or knowledge.
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Hi everyone,
In engineering design, there are usually only a few data points or low order moments, so it is meaningful to fit a relatively accurate probability density function to guide engineering design. What are the methods of fitting probability density functions through small amounts of data or low order statistical moments?
Best regards
Tao Wang
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Good explanation is performed by Chao Dang,
Best regards
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Dear colleagues,
I would appreciate if you give comments on the following question.
Best regards
Ali Taghavi
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George Stoica
Thank you for the suggestion, George
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We need to prepare a weighted average multi-model ensemble of projected future daily precipitation by assigning weights to individual CMIP6 models based on past performance. For this purpose, We want to use Bayesian Model Averaging. Since the distribution of precipitation is highly skewed with large number of zeros in it, a mixed (discrete-gamma) distribution is preferred as the conditional PDF as per Sloughter et al., (2007).
Considering 'y' as the reference (observed ) data and 'fk' as the modelled data of kth model,
The conditional PDF consists of two parts. The first part estimates P(y=0|fk) using a logistic regression model. The second part consists the following the term P(y>0|fk)*g(y|fk).
Since the computation of P(y>0|fk) is not mentioned in the referred manuscript, If I can compute P(y=0|fk), Can I compute P(y>0|fk) as 1-P(y=0|fk) in this case?
If not, Can someone help in computing P(y>0|fk)?
You can find the the referred paper here https://doi.org/10.1175/MWR3441.1
Thanks
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Yes. You can proceed with that formula as you deal with Precipitation data, which contains only non-negative values. according to axioms of probability P(y≠0|fk)=1-P(y=0|fk).
You can find a worked example in the book titled “Statistical Methods in Hydrology and Hydroclimatology(DOI: 10.1007/978-981-10-8779-0)”
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It is said Bell's inequality is a consequence of probability theory, which has nothing to do with quantum or not quantum. There are many papers discuss this issue, but I don't know which one is the original? Where can I find such material? Thanks.
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Attached for your kind perusal.
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The birth and death probabilities are p_i and q_i respectively and (1-(p_i+q_i)) is the probability for no change in the process. zero ({0}) is an absorbing state and sate space is {0,1,2, ...}. What are the conditions for {0} to be recurrence (positive or null)? Is the set {1,2,3,...} transient? What we can say about duration of process until absorption and stationary distribution if it exists and etc?
Every comment is appreciated.
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There is no logical (reasonable) condition that {0} is not absorbing, so it is always a recurrence state. {1,2,...} is always transient.
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I created a new test for uniformity, but so far, I've had no luck finding its critical values analytically, I could only obtain them by Monte Carlo simulation. What's worse is that histograms show that the null distribution does not approach normal distribution even at large n, so I cannot approximate it with mean and standard deviation.
Is there any sort of "standard procedure" for deriving null distribution of a test statistic? Or at least approximating it with an analytical expression?
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Dear Igor Yegin
You might start by looking at how the distributions for existing tests of uniformity are obtained. This seems to be via theory for Weiner Processes (also called Empirical Processses). A book that contains all this is the following:
Empirical Processes with Applications to Statistics
G.R. Shorak & J.A. Wellner
Wiley , 1986 ISBN 0-471-86725-X
(there might be possibly later editions)
This paper:
Conference Paper A review of the properties of tests for uniformity
notes a transformed version of the Sherman statistic that supposedly has improved convergence to normality. It looks to have been obtained empirically, but the papers referenced for this may make it clear.
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How to calculate the sum and the subtraction of many random variables that follow exponential distributions and have different parameters ?
(The value of Lambda is different for all or some variables).
example :
L(t) = f(t) + g(t) - h(t)
with
f(t) = a.expo(-a.t)
g(t) = b.expo(-b.t)
h(t) = c.expo(-c.t)
st:
a = Lambda_1
b = Lambda_2
c = Lambda_3.
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(continued)
In case of more terms (all with different means m_j>0, j=1,2,...,n) the formulas are as follows (ti replaced by -s)
ch.f.(X_1+X_2+...+X_n)(t) = 1/ [ (1+m_1 s) (1 + m_2 s) ... (1 + m_n s)]
= \sum_{j=1}^n A_j / (1+m_j s),
where A_j = \prod_{k\ne j} [ 1 - m_k / m_j]^{-1}
Therefore, in such cases the density of the sum is equal to
\sum_{j=1}^n A_j / m_j \exp( - x/m_j ), for x>0.
If X_j in the sum is preceded by sign -, then the first two formulas remain valid after replacing m_j by - m_j. The last requires replacing the exponential density for positive variable by the opposite one.
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In discussing Quantum Mechanics (QM), I shall restrict myself here to Schroedinger's Non-Relativistic Wave Mechanics (WM), as Dirac showed (in his 1930 text) [using Hilbert State Vectors] that Heisenberg's Matrix Mechanics (MM) was simply mathematically equivalent.
WM was invented in 1925 when Schroedinger adopted de Broglie's radical proposal that a quantum particle, like an electron, could "have" both contradictory point particle properties (like momentum, P) and wave properties, like a wave-length or wave-number K) by: K = h P; where h is Planck's constant (smuggling in quantization). Next he ASSUMED that a free electron could be represented as a spherical wave described by the Wave Equation. Then, he "joined the QM Club" by restricting his math approach to an isolated hydrogen atom, with its one orbital electron moving around the single proton (each with only one electron charge,e) at a spatial separation r at time t (i.e. x;t). He then linearized out the time by assuming a harmonic form: Exp{i w t) along with Einstein's EM frequency (photon) rule: E = h w. This gave him his famous Wave Equation [using W instead of Greek letter, psi]: H W = E W where H was the classical particle Hamiltonian H =K+U with K the kinetic energy [K= p2/2m] and U the Coulomb potential energy [U = e2/r]. Replacing the quadratic momentum term gave the Laplacian in 3D spherical polar co-ordinates [r, theta, phi]. He then remembered this resembled the 19th century oscillating sphere model with its known complete (infinite series) solution for n=1 to N=infinity for W=Y(l:cos theta) exp[i m phi] introducing the integer parameters l [angular momentum] and m [rotation around the Z axis]. By assuming the math solution is separable, he was left with the linear radial equation that could be solved [with difficulty] but approximated to Bohr's 1913 2D circular [planetary] model E values.
The "TRICK" was to isolate out from the infinite sums, all terms that only included EACH of the finite n terms [measured from n=1 to 6]. This was Dirac's key to match the nth wave function W(n:x,t) with his own Hilbert ket vector: W(n:x,t) = |n, x, t>.
So, I maintain that QM has failed to map its mathematics to a SINGLE hydrogen atom [the physical assumptions used therein] but to the full [almost infinite] collection of atoms present in real experiments. This then results in multiple epistemological nonsense such as Born's probability theory, wave function collapse and the multiverse theory.
This is NOT needed IF we reject the Continuum Hypothesis [imported from Classical Mechanics] and stick to finite difference mathematics.
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QM, like CM, imagines time as a "Fourth Dimension" (orthogonal to the 3 space directions). Physics then adopts Newton's Timeless calculus to write a set of equations that are VALID across all space at the same single time [t].
Apart from great simplifications, what is the physical justification for this model?
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In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process BH(t) on [0, T], that starts at zero, has expectation zero for all t in [0, T],
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Thanks for your useful answers.
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I am working in statistical seismology and we are running into a HIGHLY controversial topic. What can we say about the largest possible event (earthquake) that could happen in an area based on data? We make estimates, but what reliability do these estimates carry? There are epistemic and random uncertainties involved. There are many theoretical estimators for this quantity but many scientist doubt that they are of any practical value. I do not believe we seismologists are qualified to do more than "rambling" about the problem and I think some input from philosophers would be extremely enlightening.
I refer to papers:
Pisarenko VF (1991). Statistical evaluation of maximum possible magnitude. Izvestiya Earth Phys 27:757–763
Zöller, G. & Holschneider, M. (2016). The Maximum Possible and the Maximum Expected
Earthquake Magnitude for Production-Induced Earthquakes at the Gas Field in Groningen, The
Netherlands. Bull. Seismol. Soc. Am. 106, 2917-2921.
Zöller, G. (2017) Comment on “Estimation of Earthquake Hazard Parameters from Incomplete Data
Files. Part III. Incorporation of Uncertainty of Earthquake‐ Occurrence Model” by Andrzej
Kijko, Ansie Smit, and Markvard A. Sellevoll. Bull. Seismol. Soc. Am. 107: 1975-1978.
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and Albania ...
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It is known that the FPE gives the time evolution of the probability density function of the stochastic differential equation.
I could not see any reference that relates the PDF obtain by the FPE with trajectories of the SDE.
for instance, consider the solution of corresponding FPE of an SDE converges to pdf=\delta{x0} asymptotically in time.
does it mean that all the trajectories of the SDE will converge to x0 asymptotically in time?
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The Fokker-Plank equation can be treated as a so-called forward Kolmogorov equation for a certain diffusion process.
To derive a stochastic equation for this diffusion process it is very useful if you know a generator of this process. Finally, to find out a form of the generator you have to consider a PDE, dual to the Fokker-Plank equation which is called the backward Kolmogorov equation. The elliptic operator in the backward Kolmogorov equation coincides with the generator of the required disffusion process. Let me give you an example.
Assume that you consider the Cauchy problem for the Fokker-Plank type equation
u_t=Lu, u(0,x)=u_0(x),
where Lu(t,x)=[A^2(x)u(t,x)]_{xx}-[a(x)u(t,x)]_x.
The dual equation is h_t+L^*h=0, where L^*h= A^2(x)h_{xx}+a(x)h_x.
As a result the required diffusion process x(t) satisfies the SDE
dx(t)=a(x(t))dt+A(x(t))dw(t), x(0)= \xi,
where w(t) is a Wiener process and \xi is a random variable independent on w(t) with the distribution density u_0(x).
You may see the book Bogachev V.I., Krylov N.V., Röckner M., Shaposhnikov S.V. "Fokker-Planck-Kolmogorov equations"
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Dear all,
I measured a variable that takes values between 0 and 0.1 (with a minimum of 0.00053). This variable will be used in a regression analysis, but it has values of skewness and kurtosis of 3.8 and 14.3, respectively, hence requiring a transformation in order to reduce those values.
I first thought about a log transformation. However, in this way, the resulting values of the variable will be negative, and I would avoid this. Another option is multiplying all values for 1,000 and then use a log transformation. But, how can I justify this choice to referees?
Have you ever experienced this problem? How have you solved it?
Thank you for your attention to this matter.
Best
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Can i delete some of variables which has more than ±2 skewness and kurtosis to get better scores?
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suggest with probable theories and examples
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Difficult to define, this concept of spiritual intelligence.
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I Will be more than happy if somebody help me in this case. Does it has an specific function in R? or we should utilize quantile -copula methods...? or other???
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Consider the following cases, where i have put my understanding
Notation - 0+ = tending to zero
{b} = singleton set
  1. lim (n-->infinity) (1/n) = 0+
  2. lim (n-->infinity) (n/n) = lim (n-->infinity) (1) = 1
  3. Step 2 can also be looked as lim (n-->infinity) ((1/n)/(1/n)) = 0+/0+= 1 (Here both 0+ are same and they are not exact 0)
  4. lim (n-->infinity) (n2/n) = infinity
  5. step 4 can also be viewed as lim (n-->infinity) ((1/n)/(1/n2)) = 0+/0+= infinity (here both 0+ are not same and one 0+ is like infinite times the other. Which is again a conclusion that 1/n or 1/n2 with limit n goes to infinity is exact zero)
Now the real question is this from probability theory or set theory.
I found this description of singleton as
{b} = infinite intersection over 'n' of (b-1/n , b+1/n]
but according to my understanding(as above), it still should represent a range of real number and not single point. For that intersection to result in a point, 1/n should be exact zero.
These two descriptions, one from probability theory and other from calculus doesn't seem to agree to each other according to my understanding.
Can you please tell where i am doing wrong ?
I might have used some terminologies carelessly, but i hope you got the point what i am trying to ask.
Thanks
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Many of the engineer's face the same problem about the exact meaning of the concept (Limit of sequence, limit f(x), etc, ). This because the short methods for computations they learned in calculus, without focusing on the analysis of the method. So, you need to read more about REAL ANALYSIS, in particular, the Archimedes Property.
In fact, the two concepts in your question are the same.
Using Lim(1/n)=0 ,we obtain the infinite ∩(b - 1/n, b+1/n ] ={ b}.
To see this,
Let x ∈ ∩(b - 1/n, b+1/n ] for any n, then
b - 1/n < x ≤ b + 1/n for any n.
As n goes to infinity (1/n) goes to zero, we obtain
b - 0 < x ≤ b + 0 hence, x = b the only possible value.
we deduce that ∩(b - 1/n, b+1/n ] ={ b}.
You may ask, are there another point a which is very close to b
and belongs to the intersection?
Assume that b - (1/n) < a ≤ b + (1/n) for any n,
- (1/n) < a - b ≤ (1/n) ( by subtracting b )
as n goes to infinity, we obtain 0 < a - b ≤ 0,
provides a = b.
Therefore, ∩(b - 1/n, b+1/n ] = { b}.
Best regards
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If X is an asymptotic normal variate then we can find the asymptotic distribution of h(X) by using delta method if h(X) \in R. But if h(X) is not Real valued function (e.g., h(X) could be a positive function), what is the asymptotic distribution of h(X)?
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I'm having a problem with one of my works. I need to use the central limit theorem, but I first need to prove that my variables are weakly dependent. Does someone have an example that I can use as a base for my work? Thanks in advance.
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The question is unclear. Does "I need to prove that my variables are weakly dependent" mean that you want to prove that the variables are:
(a) not independent;
(b) not strongly dependent.?
In the context of the central limit theorem,there needs to be some idea of an increasing data-set. What is this?
In some types of practical application, it may not be possible to "prove" anything in the sense of a mathematical proof starting from some standard model (that then itself needs to be justified), but you be able to justify (from the practical context and experience in that context) making the direct assumption that weak dependence applies. I hesitate to point you to my own publication , as it may not be relevant to you,
Again, when you say "I need to use the central limit theorem", are you wanting to use it to get a formal asymptotic distributional result without actually wanting to apply it , or do you need a practically useful approximate result? If the latter, it may be useful to concentrate on the variance, and its asymptotic behaviour, as this should give some clue about the effects of various levels of dependence (where, for studying the variance, interest is limited to the correlation properties only).
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according to the probability theory,
suppose that we calculate the experimental probability of students who prefer mathematics and it was .70%from a sample of 20 students (14/20), is that correct to use these percentage (70% to calculate the probability of prefer mathematics in case of applying the same survey on a sample of 200 students?
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14/20 is a proportion or percentage. It's a relative frequency, not a probability. One may (!) use the relative frequeny as an estimate for the probability, that a student sampled under similar conditions will prefer mathematics (and this also only in the case that there is a countable set of possible alternatives).
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I've been thinking about this topic for a while. I admit that I still need to do more work to fully understand the full implications of the problem, but suggests that under certain conditions, Bayesian inference may have pathological results. Does it matter for science? Can we just avoid theories that generate those pathologies? If not, what can we do about it?
I have provided a more detailed commentary here:
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My practical experience suggests that that different priors do not usually make much difference unless there is very liitle information in the data. indded different results could be seen as a feature and not a problem so that you could do a sesnitivity analysis with different priors. There is a nice eaxmple of this in Clayton and Hills Statsitical Methods in epidemiology
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Dear researchers , I'm a student in Master 1 (EDP) , and am a beginner in research , I have one international paper entitled " A new special function and it's application in probability " , I want people here to Give me comments to improve that research for the futur contribution in mathematics ? , Now I want theorist in probability and numerical analyis to give us any constrictive opinion about that research in all needed sides , For checking that paper via the journal webpage , just to check this link , Thanks som much for any comments or any kind of help.
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If you have developed a new probability density function, this first suggests that you have taken into account a new phenomenon that needs to be estimated and that is not adaptable to classical probability laws.
If so, how would it facilitate a way of life, phenomena that are difficult to measure with certainty in the present, and that merits the trouble of evaluating its chances of being realized?
Or, what information or role does it assume in other disciplines: in medicine, physics, statistics, demography, risks, etc.?
Science is a kind of molecule, that is to say, which attaches, although it is specific, to other external notions. In this case, how to adapt this theory in others to facilitate its application in the real world.
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Most multivariate techniques, such as Linear Discriminant Analysis (LDA), Factor Analysis, MANOVA and Multivariate Regression are based on an assumption of multivariate normality. On occasion when you report such an application, the Editor or Reviewer will challenge whether you have established the applicability of that assumption to your data. How does one do that and what sample size do you need relative to the number of variables? You can check for certain properties of the multivariate normal distribution, such as marginal normality, linearity of all relationships between variables and normality of all linear combinations. But is there a definitive test or battery of tests?
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you can test multivariate normality by assessing the multivariate skewness and kurtosis through the software in the following link
for more information about multivariate normality please refer to the attached papers
  • Cain, M. K., Zhang, Z., & Yuan, K. H. (2017). Univariate and multivariate skewness and kurtosis for measuring nonnormality: Prevalence, influence and estimation. Behavior research methods, 49(5), 1716-1735.
  • Ramayah, T., Yeap, J. A., Ahmad, N. H., Halim, H. A., & Rahman, S. A. (2017). Testing a confirmatory model of facebook usage in smartpls using consistent PLS. International Journal of Business and Innovation, 3(2), 1-14.
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If f(t) represents the probability density of failure rate, then how it it possible that f(t) will follow exponential distribution whereas the failure rate is constant?
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Dear Parag Sen,
your problem concerns the relationship between Poisson distribution and exponential distribution - namely:
If the random variable X represents the number of errors (system failures) in a given time period and has the Poisson distribution, then the intervals between every two consecutive errors have the exponential distribution.
For example, see Wikipedia:
If for every t > 0 the number of arrivals in the time interval [0, t] follows the Poisson distribution with mean value λt, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1/λ.
In general, the exponential distribution describes the distribution of time intervals between every two subsequent Poisson events.
The answer to your question can be found at the following addresses:
Best regards
Anatol Badach
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I would like to find the probability distribution of log[U/(1-U)] when U~u(0,1). How to derive this?
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I do not quite agree with the answer of Wulf Rehder. He has computed only the cumulative distribution function(CDF). The question wants the probability distribution function(PDF), so we need to take a derivative over the CDF.
In the first case, PDF = 1/(1+x)^2 = (1-u)^2
In the second case, PDF = e^x/(1+e^x)^2 = u(1-u)
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Suppose we have n samples (x1, x2, …, xn) independently taken from a normal distribution, where known variance σ2 and unknown mean μ.
Considering non-informative prior distributions, the posterior distribution of the mean p(μ/D) follows normal distribution with μn and σn2, where μn is the sample mean of the n samples (i.e., μn=(x1+x2+…+xn)/n), σn2 is σ2/n, and D = {x1, x2, …, xn} (i.e., p(μ/D) ~ N(μn, σ2/n)).
Let the new data D’ be {x1, x2, …, xn, x1new, x2new, …, xknew}. That is, we take additional k (k<n) samples independently from the original distribution N(μ, σ2). However, before taking the additional samples, we can know the posterior predictive distribution for the additional sample. According to Bayesian statistics, the posterior predictive distribution p(xnew/D) follows normal distribution with μn and σn2+ σ2 (i.e., p(xnew/D) ~ N(μn, σ2/n+ σ2)). Namely, the variance becomes higher to reflect the uncertainty of μ. So far, this is what I know.
My question is, if we know p(xnew/D) for the additional samples, can we predict the posterior distribution p(μ/D’) before taking the additional k samples? I think that p(μ/D’) seems to be calculated based on p(xnew/D), but I have not gotten the answer yet. So, I need help. Please borrow your wisdom. Thanks in advance.
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I am not sure that I am right but lets add something to the discussion.
As likelihod is the joint prob (prodct of pdfs)of the sample and also prior/posterior predictive is the prob fnctn of xnew, therefore, to me, the liklihood fntn of x and xnew is the product of likelihood of x and prior/posterior predictive of xnew. U have the likelihood fnctn u can derive the posterior p(mu/D’).
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At the end of the nineteenth century, many researchers concentrated on various alternative methods based on the theory of infinite series. These methods have been combined under a single heading which is called \textit{Summability Methods}.  In recent years, these methods  have been used in approximation by linear positive operators. Also, in connection with the concept of statistical convergence and statistical summability, many useful developments have been used in various contexts, for example, approximation theory, probability theory, quantum mechanics, analytic continuation, Fourier analysis, the behaviors of dynamical systems, the theory of orthogonal series, and fixed point theory
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I think that it is almost completely correct and useful (especially for young researchers), each of the previous Responders' replies to the question of Professor Ugur Kadak: "What do you think about the future of summability and its applications?".
With a cordial greetings,
Jasmina Fatkić
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I have an upcoming exam, with 8 questions that may cover 10 fields. Suppose the pass criteria is 70% then how many fields i have to study to clear the exam?
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Following
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Slutsky theorem is commonly used to prove the consistency of estimators in Econometrics.
The theorem is stated as:
For a continuous function g(X_k) that is not a function of k,
plim g(X_k) = g (plim X_k)
where X_k is the sequence of random variables.
Could anyone suggest any literature on how to prove this theorem?
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The question of whether Quantum Mechanics is a complete science had sparked a historic debate led by Albert Einstein on one side and Niels Bohr on the other side. It is interesting that quantum physicists from the school of Copenhagen had to resort to philosophical arguments to defend the soundness of quantum mechanics in terms of its ability to faithfully interpret dynamic systems. The fuzziness of the central notion of the quantum wavefunction seems to have never been resolved to this day, a problem that prompted Richard Feynman in no small part to assert that “Nobody understands quantum mechanics”. I offer the view that the very mathematical tool at work in QM, the Theory of Probability (ToP), might be the first element responsible for the weaknesses of QM as a science. In Chapter 7 of the Title Quanto-Geometry: Overture of Cosmic Consciousness or Universal Knowledge for All, I discuss its limits and show the necessary extensions required for causal categories of interpretation in ToP, thus leading to completeness of QM. Downloadable here:
What do you think? Is QM obscure in its soul as formulated or are its limits attributable to Statistical Theory? Do you think the proposed extensions contribute any further intelligibility at all? What aspect of QM do you find particularly challenging?
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Hello Vikram.
Wish you well equally in the needed works of unblocking the avenues of theoretical physics.
Cordially.
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This is an example in Durrett's book "Probability theory: theory and examples", it's about the coupling time in Markov chains, but I can't see the reason behind it.
The trick is played by two persons A and B. A writes 100 digits from 0-9 randomly, B choose one of the first 10 numbers and does not tell A. If B has chosen 7,say, he counts 7 places along the list, notes the digits at the location, and continue the process. If the digit is 0 he counts 10. A possible sequence is underlined in the list:
3 4 7 8 2 3 7 5 6 1 6 4 6 5 7 8 3 1 5 3 0 7 9  2  3 .........
The trick is that, without knowing B's first digit, A can point to B's final stopping location. He just starts the process from any one of the first 10 places, and conclude that he's stopping location is the same as B's. The probability of making an error is less than 3%.
I'm puzzled by the reasoning behind the example, can anyone explain it to me ?
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The key point is that if A and B ever use the same position in their respective sequences, then from that point on their sequences are identical.  The reason that the trick works is that there is a fairly high probability that this will happen. 
Coupling in Markov chains works in a similar way.
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I saw a claim in some paper without proof. Because the formula is complicated, I wrote it in the attached file. I can not prove one part of the assertion. Please tell me how to prove it if you can understand.
Thank you
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Dear  Alejandro Santoyo
Thanks for your reply.
The attached file is my reply.
Best regard.
Masahiro Fujimoto
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Extinction probability and expected number of progeny are to be calculated at the end
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Choose the branch at random, depending on its probability, each time the branch comes up.
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In order to get a homogeneous population by inspecting two conditions and filtering  the entire population (all possible members) according these two conditions, then used the all remaining filtered members in the research, Is it still population? or it is a sample ( what is called?).
working on mathematical equation by adding other part to it then find the solution and applying it on the real world. can we generalize its result to other real world?
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Rula -
I am not sure I understand your process, but if your 'sample' is really just a census of a special part of your population, then you can get descriptive statistics on it, but you cannot do inference to the entire population from it. 
You might find the following instructive and entertaining.  I think it is quite good.
Ken Brewer's Waksberg Award article: 
 Brewer, K.R.W. (2014), “Three controversies in the history of survey sampling,” Survey Methodology,
(December 2013/January 2014), Vol 39, No 2, pp. 249-262. Statistics Canada, Catalogue No. 12-001-X.
Cheers - Jim
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Is there any other ideas than "probability theory" to derive the intervals from the responses in intervall-type Type-2 Takagi-Sugeno Systems?
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Dear Prof. Mohamed-Mourad Lafifi, I would like to thank you for your valuble information and time.
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There are traditional ways of getting the cut-off value that maximize the sensitivity and specificity of a given test from the ROC curve. But I think those ways posses some arbitrariness in doing the job. How about this;
Can we use Bayes formula to calculate the value of the variable which equates the probability of the disease to the probability of the control, assuming that both classes are normally distributed? This should be the ideal cut-off, without considering any factors related to the cost,,,etc that can affect the conventional method. So from a set of observations, we can use referential statistics to calculate the mean and the standard deviation of both classes, if not already known. Then by letting the cut-off to be the value that makes the conditional probability of the disease given the value equal to the conditional probability of the control given the value, we can set the cut-off value as a function of the difference between the two means and a function of SD. Moreover, if we consider the prior probabilities of both classes not to be equal to each other, we can even see how the cut-off values moves toward one side of the variable scale in a very natural way.  This can also help if we have more than one variable and in this case it would work similar to discriminant analysis methods.
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Bayes is still considered one of the highly popular  methods, although using support vector machines provides better  support  for fixing cut off values in a more natural way then delimiting them through baye's classifier. 
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source modeling
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Some comments:
1. If  m  is a not random constant, than one can use the name  Poisson pd on the ordered set {0=0^m, 1=1^m, 2^m, 3^m, 4^m,. . . .}
2. The name  log-Poisson should be accordingly preferred for    e^x, instead;  indeed, y=e^x  is of log-normal distribution if x is normal.
3. If   m  is random, no particular names can be assignedalso, not any particular  distribution can be implied without information on the joint pd of  n  and  m
4. For the limit pd of sums like  a logx+b logx+c logx+d logx +...,  the CLT is applicable in very special circumstnces only, which in particular  have to take into account non-zero correlation between  the coefficients a,b,c,d,...and the rv  x in such a way, that the terms are iid;  this is due to the fact, that if  a,b,c,d,... are iid, then   a logx, b logx, c logx, d logx , ...  are not independent
Further comments are in prepartion:)
Best regards, Joachim
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I did analyzed these counts (about 100 C/S) by poisson function and gaussian function and i observed that, nearly there is no difference between using of a gaussian and poisson functions. 
Although, i know that: 1- gaussian function is used if the count rates are greater than 20 count 2- poisson function is used if the count rates are smaller than 20 count.
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@Deepak
Good catch. @Mahmoud said analyzed counts, but discussed count rates. One must be very careful with count rates. It is always better to do the analysis in counts. Many count rate formulas are incorrect or used incorrectly. The problem is combining count rates taken over different time periods.
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Bottleneck analysis software offers three models (IAM, TPM and SMM) and three tests (Sign, Standardized difference and Wilcoxon). IAM is known for use with allozymes, and TPM and SMM with microsatellites. The latter models under different tests above show presence of bottleneck while not so under another test.
For example; 
SIGN TEST
Assumptions: all loci fit T.P.M., mutation-drift equilibrium.
Expected number of loci with heterozygosity excess: 18.06
8 loci with heterozygosity deficiency and 22 loci with heterozygosity excess.
Probability: 0.09738
Assumptions: all loci fit S.M.M., mutation-drift equilibrium.
Expected number of loci with heterozygosity excess: 17.80
13 loci with heterozygosity deficiency and 17 loci with heterozygosity excess.
Probability: 0.45129
______________________________________________________________
STANDARDIZED DIFFERENCES TEST
Assumptions: all loci fit T.P.M., mutation-drift equilibrium.
T2: 2.072 Probability: 0.01913
Assumptions: all loci fit S.M.M., mutation-drift equilibrium.
T2: -0.802 Probability: 0.21140
_____________________________________________________________
WILCOXON TEST
Assumptions: all loci fit T.P.M., mutation-drift equilibrium.
Probability (one tail for H deficiency): 0.99017
Probability (one tail for H excess): 0.01042
Probability (two tails for H excess or deficiency): 0.02085
Assumptions: all loci fit S.M.M., mutation-drift equilibrium.
Probability (one tail for H deficiency): 0.59608
Probability (one tail for H excess): 0.41179
Probability (two tails for H excess or deficiency): 0.82358
___________________________________________________________
MODE-SHIFT
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
| 0.410 | 0.312 | 0.195 | 0.044 | 0.029 | 0.000 | 0.010 | 0.000 | 0.000 | 0.000 |
When different models under different tests produces different results on presence bottleneck as in Standardized difference and Wilcoxon test above. What is the bottom line of drawing the conclusion on presence or absence of bottleneck in a population? 
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Dear Dragos, Thanks for the article. Interesting and helpful indeed. 
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Dear all,
I am searching a result concerning the weak convergence of signed measures defined on the Borel algebra of a given metric space. The result that i find so far is established for weak convergence of probability measures (please see Portmanteau theorem page 11; book attached).
Does the same result apply for signed measures? If not, what could be the possible differences in the equivalent conditions?
(providing me with good references would be appreciated)
Sincerely,
Nathalie
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Dear All Followers of this question,
let me give answer to my subproblem by presenting a counterexample, where $\delta(a)$  stands for the p.d. concentrated at $a$, for $a \in R$:
For  $n \in Z_0=\{...,-2,-1, 1,2,3, . . . \}$  let  the coefficients be
$b_n : =  sign(n) \sqrt{ \pi |n|}$ and let the signed measures be
$Q_n := ( -\delta(b_{-2 n-1} +\delta(b_{-2 n} + \delta(b_{2 n} - \delta(b_{2 n+1} $.
Their characteristic functions equal  
$ f_n(t)  = 2 [  \cos(t \sqrt{2n\pi} - \cos(t \sqrt{(2n+1)\pi} ] $
By elementary trigonometry, $f_n(t) \to 0$,  for all $t \in R$.
On the other hand the integrals of the obviously continuous and bounded function $\cos(x^2), x \in \R$ with respect to $Q_n$ are all equal to
 $ \int_R \, cos(x^2) \,  dQ_n(x) = 2  [ \cos ((b_{2n})^2) -  \cos ((b_{2n+1})^2) $,
which equals 4, for every  $n \in  N$. Thus the measures $Q_n$ are not weakly convergent to the zero measure, although absolute variations of the measures are bounded (equal 4).
Thus, thank you Natalie for your question! Without it we could  miss this special feature of signed measures.
Joachim
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Let's say the initial population in NSGA ii is  i.i.d and uniformly distributed. Has anyone done research about what we can say about the distribution after k iterations in NSGA ii? The individuals are surely no longer i.i.d but are there asymptotic results for large populations sizes?
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Dear Ernst,
You can utilize the following codes if you use MATLAB as follows:
before merging the population at the end of algorithm, please type "Stop time;"
and save the initial population in another Pop, e.g. Pop2.
Yours,
Niknamfar
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Dear researchers,
How can be calculated the failure probability for a component with λ=1*E-6 failure per hour and a month (τ=1 month) proof test interval at one year؟
Please see the following files.
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Dear Dr Limon,
Thanks.
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Which equation (14 or 15) in the paper (attached in the link) gives us the average outage probability?
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Outage probability is the probability that a system does not function as per the expectations, such as the probability when the instantaneous SNR falls below a threshold SNR. But if the SNR considered is received SNR, then that received SNR which depends on the wireless channel could be considered random. So, we need to average out the received SNRs for various channel realizations. Hope this helps. 
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I have read that a drawback with Edgeworth series expansion is that "... they can be inaccurate, especially in the tails, due to mainly two reasons: (1) They are obtained under a Taylor series around the mean. (2) They guarantee (asymptotically) an absolute error, not a relative one. This is an issue when one wants to approximate very small quantities, for which the absolute error might be small, but the relative error important."
So my resulting question is if there are any attractive alternative ways of approximating stochastic variables with some corresponding method that still is useful in the tails of the distribution, and does not (for example) result in negative probabilities (which is mentioned as another drawback with the approach).
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Another criticism of the Edgeworth (and Gram-Charlier) series is that they do not always (usually?) converge--they are similar to asymptotic expansions).
Instead of expanding the density, you could expand a function that transforms it to normality. The Cornish-Fisher expansions (for such a transformation and its inverse) achieve this. These are based on the Edgeworth expansion. They might also fail to converge but still give approximations.
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Let {A_n} be a sequence of sets, limsup A_n=, liminf A_n and lim A_n are defined in probability theory.
I would like to know who introduced this notions and find some papers on this context!
Thanks so much!
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Hello,
I think there is not a unique answer to your question. However :
- Fréchet in his thesis in 1906 introduced the bases for topology over general sets and went on with several works on abstract sets in the following years
- the liminf definition may be due to Borel in his 1909 paper introducing the strong law of large numbers by means of (today's name of course) Borel-Cantelli lemma expressing that only a finite number of events in a sequence take place.
Sincerely,
Laurent
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If I have two independent variables (time series) for two different sources(let's say from Observation and from Model).
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Thank You Juan and Shameek.
Juan your chapter may help in my work . Actually I want to use this technique for rainfall forecasting for monthly and Seasonally.
Thank you again 
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I would like to study the problem of insolvency in the insurance companies and what I thought about is classical probability models where I can examine the probability of the ultimate ruin ?
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Bear in mind, however, that in the real world ruin often occurs not because a threshold of extreme value was exceeded, but because the model used to estimate the capital requirements was insufficient.  For example, a model may assume that risks in different regions or different sectors are independent and can be used to hedge each other, but situations will always exist where the risks are correlated, with one event then resulting in losses over many sectors or regions.
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significance of gender as dummy variable
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Additionally, although this might be practically unimportant in your case, if you have no other IV, regressing the outcome variable only on gender provides you with the average outcome of the base category of your gender variable with the intercept of the model, and the difference to that intercept with the regression coefficient of your gender variable.
y = 3  - 1*Gender would mean that for your base category of gender (Gender = 0), the average of your outcome variable is 3, and for Gender = 1, the average of the outcome variable is 2 (3 - 1). Regression tells you whether this difference is significant.
If you have other control variables, this would be a bit more complex. See here for a very easy and basic introduction: http://www.theanalysisfactor.com/interpreting-regression-coefficients/
Hope this helps.
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Hi everybody,
I'm going to apply multiDE, an R package for detection of Differentially Expressed Genes in multiple treatment conditions, on some RNA-seq time series data... (I want to assume each time point as a treatment condition)
Let's assume Yidg denotes the normalized read counts for sample "i", in condition "d" for gene "g".
We also assume that Yidg marginally follows the negative binomial distribution with expectation "μdg" and dispersion parameter "ϕg" (i.e., the variance of Yidg is μdg+ϕgμ2dg).
The statistical methodology behind this package is a two factorial log linear model : logμdg = μ + αd + βg + γdg = μ + αd + βg + UdVg,
where μ is the grand mean, αd is the main effect for condition d, βg is the main effect for gene g, and γdg:=UdVg is the interaction effect between gene g and condition d.
My professor has asked me to estimate the main effect for condition (α), the main effect for gene (β) and the effect of interaction between gene and condition (γ). While the package can only show "Ud" in its output...
I'm in grave need of help to find out how I can estimate those effects please...
My main problem is I don't know how I can calculate μdg. Maybe if I can calculate it, then applying a regression strategy would be helpful to estimate those effects...
here it is the link to the full paper: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4917940/
Thanks in advance
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To reply in detail to your question would require that I download your paper and examine in detail what you would like to do.
In general the application of log-linear models is well know and excellent books are available, so just chose one of them  and pursue your research.
But, it seems, in my experience see in my directory the paper
Article: Solving large protein secondary structure classification problems by a nonlinear complementarity algorithm with {0, 1} variables
C. Cifarelli · G. Patrizi
Full-text Article · Feb 2007  and the  Optimization Methods and Software  that you need a log-linear model to estimate categorical effects such as the ones that you mention.
In this case, there are three excellent books, a bit dated but surely the best, written by Shelby Haberman, especially the 2 volume work entitled " Analysis of Qualitative Data" , which your University Library can borrow from my department if it is not available in your nearby university libraries.  Perhaps you can use also books on categorical analysis by Alan Agresti and there should be the necessary software also available in R.
I will look into your problem very willingly if I am given some time to study your paper supposing that this research probably concerns your thesis or a joint paper.
Looking forward to your answer,
Giacomo Patrizi
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Well! Generalized (G-) classes with more than 2 parameters are around. The questions is "the model with more than 3 shape parameters is worthy?" Many statisticians have on objection on fitting models with more than 3 parameters. Well! we all should think about it. Especially the recent work of Morad Alizadeh, Ahmed Afify and Laba Handique. The special models contain 4 or more parameters . Look what is happening with modern distribution theory! You can fit an elephant with 4 or 5 parameters. A very serious concern.
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In regression analysis we consider a large number of parameters to limited size data and nobody bothers about it.Why so much concern when a distribution contains three shape parameters?.The larger question in both cases is how successfully we can deal with the estimation problem
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Given a set of values of a variable x= 40, 25, 20, 15, 12, and 5:
The probability of exceedance of a value x=20 written as  P (x>=20)  can be got by arranging data is descending order thus giving a value of 0.5
The probability of non-exceedance of a value x=20 written as  P (x<=20) and can be got by arranging data is ascending order thus giving a value of 0.67
On checking P(x>=1) = 1 - P(x<=1) gives
0.5 = 1 - 0.67 which is not correct. Is this error creating by the estimations made using the probability formulae?
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First of all the Correct fact is: P(x>=1) = 1 - P(x<1)
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I would like to capture the dependence between returns using regime switching copulas and I'd like to know if there is any code currently available.
More in details, I would like to estimate the maximum likelihood estimates using the EM algorithm displayed in Hamilton in particular. In the framework, we consider two states of the economy, each one characterized by a certain copula and dependence parameter.
Thank you very much in advance.
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Code is available for R software. You need to edit for your specific use.
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When I use adequacy Model to fit the new probability model by use appropriate data sets.
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Are you running some sort of nonlinear optimization?
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I have considered Hydraulic conductivity (Ks) of clayey soil as random variable with log normal distribution. I have got negative mean (lambda) after the determination of measures of variation. Logically, I should not have negative mean for physical parameter Ks.. Find the attached excel document.. Kindly provide solution as soon as possible..
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Yes, it is possible to have a negative value for lognormal mean. The main purpose of using a lognormal distribution for probabilistic analysis is to have only positive values assigned to the variables (engineering properties like hydraulic conductivity). The lognormal of the variables is normally distributed however, you would be interested in assigning the corresponding normally distributed value to the hydraulic conductivity, and in that case you will happen to raise the lognormal mean to the exponent function rendering positive values only (exponent function is always positive). Hope this helps.
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Hello! I am only starting to study GRP (General Renewal Process), so sorry, if my question will be too simple -:). Main expression for GRP is A[i] - A[i-1] = q*(S[i] - S[i-1]), where A is Virtual Age, S is Real Time and q is restoration factor. It is clear (???), that this formula also proved for non-constant restoration factor q[i] – e.g., q[i] = q_CM for Corrective Maintenance and q[i] = q_PM for Preventive Maintenance. But I see some collision, if sometimes q[i] = 0 (e.g., q_PM = 0). On the one hand, q[i] = 0 means, that it is replacement (Good as New) and in this case the Aging is absent, i.e. A[i] = 0. On the other hand, according above formula, A[i] = A[i-1] and it isn't 0, because q[i-1] =/= 0. What is your opinion?
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Why we are using the (characteristics function) CHF for evaluating the (probability density function) PDF of any random variable, why not directly evaluate PDF for random variable..
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It is not necessary to use CHF .  For example if you know the distribution function (DF) then  its derivative is the PDF.
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I am working on simply supported beam, please let me know any hint or idea?
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MATLAB can help you building a PDF graph. You can do the same with Python using commonly known libraries; although there many software choices. You can read about statistics and PDF functions in the following books:
J. P. Marques de Sá, "Applied Statistics using SPSS, STATISTICA, MATLAB and R", 2007.
Andrew N. O’Connor,"Probability Distributions Used in Reliability Engineering", 2011.
Good Luck!
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If the homogeneity of regression slopes assumption for ANCOVA and levene’s test was violated (significance ), what is the alternative test for ANCOVA? I have post-test mean scores as dependent, pre-test scores as covariates, one independent variables with two treatment modes, and one moderator variables. Is it Welch’s test suitable for ANCOVA?
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It is not an alternative actually but you could run a linear regression and check for assumptions of it.
Or you could make use of a mixed model allowing for non-homogenous variances.
Another one might be going on with ANCOVA despite violations.
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My data lie in the unit interval (proportion). I assume them to be drawn from a beta distribution with parameters a and b. Are there recommendations for the choice of the priors for a and b ?
Thank you!
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If you want an uninformative prior, you may consider using Jeffrey's Prior.
Kathryn Blackmond Laskey, Bayesian Inference and Decision Theory, Unit 7: Hierarchical Bayesian Models recommends that you transform the hyperparameters of the beta distribution. Dr Laskey proposes this transformation because the two hyperparameters of the beta distribution are so closely correlated that the Gibbs sampling is inefficient. Her transformation, and the only transformation that others consider is U=alpha/(alpha+beta) and V=alpha+beta. U is the mean of the beta distribution and V is a rough measure of the precision of the beta distribution with larger values implying more precision. She suggests a uniform distribution for U and a gamma(1,20) distribution for V. She uses the rat tumour data from Gelman et al.
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How can the probability density function of a Weibull distributed random variable be closely approximated ?
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mxed exponentional distribution, or hyper- erlang, or COXIAN DISTRIBUTION 
Maybe Acycle Ph distribution is the best chioce!
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Let \a = (a_1,a_2,...) \in {0,1,2}^\N be a 3m-periodic sequence (i.e., a_{j+3m}=a_j for all j\in\N). 
ASSUME that # {1 \le r \le 3m : a_r = i}=m for i=0,1, and 2
(i.e. the 0's, 1's, and 2's appear the same number of times in each period).  
For i=1,2 define the function U^i:\N \to \N by 
U^i(x):= # of i's in the sequence \a prior to x appearances of 0
(i.e. let T(x) = min{t: \sum_{r=1}^t 1_{a_r=0} =x},
then U^i(x)=\# {1\le r \le T(x): a_r=i} ).
Let U^i_j, i=1,2 and j=1,...,m, be the function 
U_i for the j-shifted sequence \a_j := (a_j,a_{j+1},...).
Is there always some 1 <= k <= 3m so that the number of couples (i,j) for which U^i_j (k) >= k is at least half of the total number of couples, namely is at least 3m?   
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Does the numbers of trials are not related to the percentages of probabilities?
I was comparing between these two Matlab codes, and have a question about which one would have higher success rate of probabilities.
Why is the second code has higher number of probabilities than the first?