Science topic

# Probability Theory - Science topic

The analysis of random phenomena and variable, stochastic processes and modeling.

Questions related to Probability Theory

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

What’s the most common programming paradigm of no-code platforms? Why?

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.

How can probability theory and statistical modeling contribute to our understanding of phonological variation and probabilistic phonological processes?

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.

Suppose A is a set measurable in the Caratheodory sense such for n in the integers, A is a subset of R

^{n}, and function f:A->RAfter 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:- The expected value of f on each term of the pre-structure is finite
- The
*pre-structure**converges uniformly*to A - 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. - 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. - 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 R^{A}. - 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.

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.

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) 306-312Popper K R 1957 The propensity interpretation of the calculus of probability and the quantum theory, In S. KSrner (ed.),

*Observation and interpretation*. Butterworths, London.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

Dear colleagues,

I would appreciate if you give comments on the following question.

Best regards

Ali Taghavi

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

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.

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.

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?

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.

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= p^{2}/2m] and U the Coulomb potential energy [U = e^{2}/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.

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*],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.

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?

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

suggest with probable theories and examples

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???

Consider the following cases, where i have put my understanding

Notation - 0

^{+}= tending to zero {b} = singleton set

- lim (n-->infinity) (1/n) = 0
^{+} - lim (n-->infinity) (n/n) = lim (n-->infinity) (1) = 1
- 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) - lim (n-->infinity) (n
^{2}/n) = infinity - step 4 can also be viewed as lim (n-->infinity) ((1/n)/(1/n
^{2})) = 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/n^{2}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

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)?

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.

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?

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:

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.

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?

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?

I would like to find the probability distribution of log[U/(1-U)] when U~u(0,1). How to derive this?

Suppose we have n samples (x

_{1}, x_{2}, …, x_{n}) independently taken from a normal distribution, where known variance σ^{2}and unknown mean μ.Considering non-informative prior distributions, the posterior distribution of the mean p(μ/D) follows normal distribution with μ

_{n}and σ_{n}^{2}, where μ_{n}is the sample mean of the n samples (i.e., μ_{n}=(x_{1}+x_{2}+…+x_{n})/n), σ_{n}^{2}is σ^{2}/n, and D = {x_{1}, x_{2}, …, x_{n}} (i.e., p(μ/D) ~ N(μ_{n}, σ^{2}/n)).Let the new data D’ be {x

_{1}, x_{2}, …, x_{n}, x_{1}^{new}, x_{2}^{new}, …, x_{k}^{new}}. That is, we take additional k (k<n) samples independently from the original distribution N(μ, σ^{2}). 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(x^{new}/D) follows normal distribution with μ_{n}and σ_{n}^{2}+ σ^{2}(i.e., p(x^{new}/D) ~ N(μ_{n}, σ^{2}/n+ σ^{2})). 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(x

_{new}/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(x^{new}/D), but I have not gotten the answer yet. So, I need help. Please borrow your wisdom. Thanks in advance.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

**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?**

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?

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?

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 ?

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

Extinction probability and expected number of progeny are to be calculated at the end

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?

Is there any other ideas than "probability theory" to derive the intervals from the responses in intervall-type Type-2 Takagi-Sugeno Systems?

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.

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.

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?

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

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?

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.

Which equation (14 or 15) in the paper (attached in the link) gives us the average outage probability?

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).

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!

If I have two independent variables (time series) for two different sources(let's say from Observation and from Model).

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 ?

significance of gender as dummy variable

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

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.

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?

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.

When I use adequacy Model to fit the new probability model by use appropriate data sets.

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..

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?

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..

I am working on simply supported beam, please let me know any hint or idea?

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?

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!

How can the probability density function of a Weibull distributed random variable be closely approximated ?

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?

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?

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