Science topic

# Probability Theory - Science topic

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

Questions related to Probability Theory

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

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???

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

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?

**Is the canonical unit 2 standard probability simplex, t**he convex hull of the equilateral triangle in the three dimensional Cartesian plane whose vertices are (1,0,0) , (0,1,0) and (0,0,1) in euclidean coordinates, closed under all and only all convex combinations of probability vector

that is the set of all non negative triples/vectors of three real numbers that are non negative and sum to 1, ?

Do any unit probability vectors, set of three non negative three numbers at each pt, if conceveid as a probability vector space, go missing; for example

**<p1=0.3, p2=0.2, p3=0.5>**may not be an element of the domain if the probability simplex in barry-centric /probabilty coordinate s a function of p1, p2, p3 .**where y denotes p2, and z denotes p3, is not constructed appropriately?**

and the pi entries of each vector, p1, p2 p3 in <p1, p2,p3> p1+p2+p3=1 pi>=0

in the x,y,z plane where x =m=1/3 for example, denotes the set of probability vectors whose first entry is 1/3 ie < p1=1/3, p2, p3> p2+p3=2/3 p1, p2, p3>=0; p1=1/3 +p2+p3=1?

p1=1/3, the coordinates value of all vectors whose first entry is x=p1=m =1/3 ie

**Does using absolute barry-centric coordinates rule out this possibility? That vector going missing?**

where <p1=0.3, p2=0.2, p3=0.5> is the vector located at p1, p2 ,p3 in absolute barycentric coordinates.

Given that its convex hull, - it is the smallest such set such that inscribed in the equilateral such that any subset of its not closed under all convex combinations of the vertices

**(I presume that this means all and only triples of non negative pi that sum to 1 are included, and because any subset may not include the vertices etc). so that the there are no vectors with negative entries**every go missing in the domain in the , when its traditionally described in three coordinates, as the

**convex hull of three standard unit vectors**(1,0 0) (0,0,1 and (0,1,0), the equilateral triangle in Cartesian coordinates x,y,z in three dimensional euclidean spaces whose vertices are Or can this only be guaranteed by representing in this fashion.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

State dependent additivity and state independent additivity? ;

akin to more to

**cauchy additivity**versus local ko**lmorgov additivity/normalization of subjective credence/utility,**in a simplex representation of subjective probability or utility ranked by objective probability distinction? Ie in the 2 or more unit simplex (at least three atomic outcomes on each unit probability vector, finitely additive space) where every events is ranked globally within vectors and between distinct vectors by < > and especially '**='****i [resume that one is mere representability and the other unique-ness**

**the distinction between the trival**

**(1)x+y+z x,y,z mutually exclusive and exhaustive F(x)+F(y)+F(z)=1**

**(2)or F(x u y) = F(x)+F(y) xu y ie F(A V B)=F(A)+F(B)=F(A)+ F(B) A, B disjoint**

**disjoint on samevector)**

**(3)F(A)+F(AC)=1 disjoint and mutually excluisve on the same uni**t vector

and more like this or the properties below something more these

to these (3a, 3b, 3C) which are uniqueness properties

**forall x,y events in the simplex**

**(3.A) F(x+y)=F(x)+F(y) cauchy addivity(same vector or probability state, or not)**

**This needs no explaining**

**aritrarily in the simplex of interest (ie whether they are on the same vector or not)**

**or(B) x+y =z+m=F(z)+F(m) (any arbitary two or more events with teh same objective sum must have the same credence sum, same vector or not) disjoint or not (almost jensens equality)**

**or (C)F(1-x-y)+F(x)+F(y)=1 *(any arbitrary three events in the simplex, same vector or not, must to one in credence if they sum to one in objective chance)**

**(D) F(1-x)+F(x)=1 any arbitary two events whose sum is one,in chance must sum to 1 in credence same probability space,/state/vector or not**

**global symmetry (distinct from complement additivity) it applies to non disjoint events on disitnct vectors to the equalities in the rank. 'rank equalities, plus complement addivitity' gives rise to this in a two outcome system, a**

. It seems to be entailed by global modal cross world rank, so long as there at least three outcome, without use of mixtures, unions or tradeoffs. Iff ones domain is the entire simplex

that is adding up function values of sums of evenst on distinct vectors to the value of some other event on some non commutting (arguably) probability vector

F(x+y)=F(x)+F(y)

In the context of certain probabilistic and/or utility unique-ness theorems,where one takes one objective probability function and tries to show that any other probability function, given ones' constraints, must be the same function.

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 a distinction between strong or complete qualitative probability orders which are considered to be strong representation or total probability relations neither of which involve in-com parables events, use the stronger form of Scott axiom (not cases of weak, partial or intermediate agreements) and both of whose representation is considered 'strong '

of type (1)P>=Q iff P(x)>= P(y)

versus type

(2) x=y iff P(x)=P(x)

y>x iff P(y)>Pr(x) and

y<x iff P(y)<Pr(x)

The last link speaks about by worry about some total orders that only use

totallity A<=B or B<=A without a trichotomy https://antimeta.wordpress.com/category

/probability/page/3/

where they refer to:

f≥g, but we don’t know whether f>g or f≈g. S

However, as it stands, this dominance principle leaves some preference relations among actions underspecified. That is, if f and g are actions such that f strictly dominates g in some states, but they have the same (or equipreferable) outcomes in the others, then we know that f≥g, but we don’t know whether f>g or f≈g. So the axioms for a partial ordering on the outcomes, together with the dominance principle, don’t suffice to uniquely specify an induced partial ordering on the acti

.

The both uses a total order over

**totality**

**A <=B or B >=A**

**l definition of equality and anti-symmetry, A=B iff A<=B and B>=A**

**A<= B iff [A< B or A=B] iff not A>B**

**A>=B iff [A>B or A=B]iff not A<B**

**where A>B equiv B<A,and**

**A>=B equiv B<=A iff (A<B)**

**where = is an equivalence relation, symmetric, transitive and reflexive**

**<=.=> are reflexive transitive, negative transitive,complementary and total**

**, whilst <, > are irreflexive and ass-ymetric,**

**transitive**

**A<B , B<C implies A>C**

**A<B B=C implies A>C**

**A<B, A<=B implies A>C**

**and negatively transitive**

**and complementary**

**A>B iff ~A<~B**

**<|=|>, are mutually exclusive.**

and where equality s, is an equivalence class not denoting identity or in-comparability but generally equality in rank (in probability) whilst the second kind uses negatively transitive weakly connected strict weak orders,r <|=|>,

**weak connected-ness not (A=B) then A<B or A> B**

whilst the second kind uses both trichotomous strongly connected strict total orders, for <|=|>,.

**(2) trichotomoy A<B or A=B or A>B are made explicit, where the relations are mutually exclusive and exhaustive in (2(**

**(3) strong connectected. not (A=B) iff A<B or A> B, and**

**and satisfy the axioms of A>= emptyset, \Omega > emptyset , \Omega >= A**

**scotts conditions and the separability and archimedean axioms and monotone continuity if required**

In the first kind <= |>= is primitive which makes me suspect, whilst in the second <|=|> are primitive.

Please see the attached document.And whether trich-otomoy fails

in the first type, which appears a bit fuzzier yet totality holds in both case A>=B or B<=B where

What is unclear is whether there is any canonical meaning to weak orders (as opposed total pre-orders, or strict weak orders) .

In the context of qualitative probability this is sometimes seen as synonymous with a complete or total order. , as opposed to a partial order which allows for incomparable s, its generally a partial order, which allows for comparable equalities but between non identical events usually put in the same equivalence class (ie A is as probable as B, when A=B, as opposed, one and the same event, or 'who knows/ for in-comparability) Fihsburn hints at a second distinction where A may not be as likely as B, and it must be the case

not A>B and not A< B yet not A=B is possible in the second yet

A>= B or A<=B must hold

which appears to say that you can quasi -compare the events (can say that A less then or more probable, than B ,but not which of the two A<B, A=B, , that is which relation it specifically stands in

but yet one cannot say that A>B or A<B

)

and satisfy definitions

and A<=B iff A<B or A=B iff B>=A, iff ~A>=~B, where this mutually exclusive to A<B equiv ~B>~A

A>=B iff A>B or A<=B

iff iff B>=A where this mutually exclusive to A>B equiv ~B<~A

and both (1) and (2) using as a total ordering over >= |<=

(1)totalityA<= B or B<=A

(2)equality in rank and anti-symmetric biconditional A=B iff A<=B and B>=A where = is an equivalence relation, symmetric, transitive and reflexive

(2) A<=B iff A<B or A=B, A>=B iff A>B or A<=B

(3) and satisfy the criterion that >|<|>=|<=, are

complementary, A>B iff ~B<~A

transitive and negatively transitive,

where A<B iff B<A and where , =, <|> are mutually exclusive,

The difference between the two seem to be whether A>=B and A<= B is equivalent to A=B; or where in the first kind, it counts as strongly respresenting the structure even if A>=B comes out A>B because one could not specify whether A>B or A=B yet you could compare them in the sense that under <= one can say that its either less or equal in probability or more than or equal, but not precisely which of the two it is.

either some weakening of anti-symmetry of the both and the fact that the first kind use

whilst the less ambiguous orders trich-otomous orders use not (A=B) iff A<B or A> B; generally trichotomy is not considered, when it comes to using satisfying scotts axiom , in its strongest sense, for strict aggreement

and I am wondering whether the trich-otomous forms which appear to be required for real valued or strictly increasing probability functions are slightly stronger, when it comes to dense order but require a stronger form of scotts axiom, that involves <. > and not just <=.

but where in (1) these <=|>= relation is primitive and trich-otomoy is not explicit, nor is strong connected-ness whilst in (2)A neq B iff A>B or A<B

>|=|< is primitive and both

(1) totality A<= B or B<=A

(2) A<B or A=B or A>B are made explicit, where the relations are mutually exclusive and exhaustive in (2(

and (2) trichotomy hold and are modelled as strict total trichotomous orders,

as opposed to a weakly connected strict weak order, with an associated total pre-order, or what may be a total order,

, or at least are made explicit. I get the impression that the first kind as deccribed by FIshburn 1970 considers a weird relation that does not involve incomparables, and is consided total but A>=B and B<=A but one cannot that A is as likely as B, or that its fuzzy in the sense

that one can say that B is either less than or equal in probability to A, or conversely, but if B<= A one cannot /need not whether say A=B or A<B,

not A=B] iff A<B or A>B

and strongly connected in the second.

where A=B iff A<=B and B>=A in both cases

where <= is transitive , negative transitive, complementary, total, and reflexive

A>=B or B<=A

are considered complete

and

y

What is general, is meant by the orthogonality relation x⊥y

in the functional equation: orthogonal additivity below (1)

(1)∀(x,y)∈(dom(F)∩[x⊥y]):F(x+y)=F(x)+F(y)

Particularly as used in the following two references listed below (by De Zelah and Ratz, in the context of quantum spin half born rule derivations

What is general, is meant by the orthogonality relation x⊥y

denotes x is orthogonal to y.

.

(1)∀(x,y)∈(dom(F)∩[x⊥y]):F(x+y)=F(x)+F(y)

See Rätz, Jürg, On orthogonally additive mappings, Aequationes Math. 28, 35-49 (1985). ZBL0569.39006.

2.See 'Comment on `Gleason-type theorem including Qubits and pro-jective measurements: the Born rule beyond quantum physics' ", by Michael J. W. Hall ">https://www.researchgate.net/profile/Francisco_De_Zela

where x⊥y

denote: -'logical/set theoretical Disjointedness' of events, on the same basis.That is mutual exclusivity? Or - or geometrical orthogonality events . That is, on perpendicular/possibly non-commuting vectors/bases in spin 12 system? (for example, spin up x direction versus spin up ydirection)? - OR something else?

That is, within QM, in the Hilbert inner product metric, what does it mean for the frame function two events to be explicitly allowed to add as per x⊥y

, in functional equation, Orthogonal additivity.

.

Does z⊥m

denote events mean events whose amplitude modul-i squared (P(A),PR(B))=(z,m)respectively and which lie on the same basis/vector in a spin system, for example

:

A spin up on basis in x direction,P(A)=||amplitude(spin up_x)||2=z

and¬A spin down on basis in x direction;P(¬A)=||amplitude(spin down_x)||2=m

2.Versus Non commuting bases:

A spin up in x direction

andA spin down in y direction

Corresponding to the logical and geometric notions roughly, respectively

How does one distinguish this, from events that are non-commuting from events& orthogonal events in the geometric sense from events, that are orthogonal in the logical sense and disjoint events on the same basis?

Is there a distinct operator, inner product in quantum mechanics (which equals zero) which determines when the frame function probabilities of events can explicitly be assumed to add?

That is disjoint, in the sense of **Kolmogorov that is disjoint or mutually exclusive, or its analog in quantum mechanics as in (1)?

(1)F(X∪Y)=F(X)+F(Y)where in (1);X∩Y=∅,∧X,Y∈Ω,,X,Y are mutually exclusive and lie on the same basis

(1a)P(A∪AC)=1,A∪AC=Ω=⊤∧A∪AC=∅

(1.b)∀(Ai)∈Ω;where,Ai∈F,the singletons, atoms, of which are in the algebra of events F;P(∪n=|Ω|i=1Ai)=[∑i=1n=|Ω|P(Ai)]=1

where, in 1(b)[∪n=|Ω|i=1Ai]=⊤∧∀(j);j≠i⟺Ai∩Aj=∅

This being opposed to: when the events on distinct vectors, that are geometrically orthogonal or non commuting/complementary bases/distinct bases, which are not explicitly specified to add, they may happen to, as was derived for n≥3

, where this is really a form of the much stronger global ,Cauchy additivity equation (2) which relate to unique-ness , when one has (1), in addition ? (2)(2)∀(x,y)∈dom(unitsphere);F(x+y)=F(x)+F(y)

where x+y,,x,y, can be translated as ||x+y||2,||x|2,||y||2 Of these two notions, to which does orthogonal additivity relate to, is probabilism (or quantum probabilism) with some further richness properties that make it closer to (2)in a restricted form

That is the sense of being on bases that are at right angles or being at 90

degrees from one another, not simultaneously measurable such as (A),(B):

as

(A)spin-up at angle y with||spin−up−y||2=P(spin up, measured@ angle,y)12

(B)spin-up at angle x with||spin−upxy||2=P(spin up, measured@ angle,x)=12

Both seem to use the Hilbert space inner product? I presume one is for bases, and another is for events.

Does this relate to when the events commute, are on vectors at right angles from each other, or rather do commute and lie on the same basis. If the former is just standard probabilism.

I was wondering if someone knows whether Mathematica allows one to plot another 'probability function over the unit 2 simplex (it can be expressed as a function of two arguments subject to certain contrainsts function .

Where I am taking the domain to be of the function to be over vectors in the 2- standard simplex itself as it were,subject to certain contrain'ts.

Does it actually have a closed form expression as a function x,y coordinates. as a function of two arguments. I presume mathematica can allows you plot it .and optimize certain functions over tenary plot, ternary graph, triangle plot, simplex plot, G? Is that correct

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

Is the ortho-complement of a proposition, in quantum logic/probability or hilbert space, the logical comple-ment of a proposition

such as 'spin up at direction x and 'spin down in direction y'

(mutually exhaustive, on the) same prepared spin system at the as same angle, ie on the same basis) and thus one can add them the probabilities to one, as in the usual probabilistic sense to .when the hilbert space inner product is zero (or rather one or zero, the kronecker delta).

Or is this a geometric notion that that allows one to add probabilities of events together that lie on distinct bases (bases/vectors on distinct angles of the same prepared spin system). Which is stronger (closer to a restricted form of cauchy additivity)

the on the very same vector in an orth-onormal basis; the very same basis.

, in the sense that they are disjoint A\cap B emptyset , and mutually exclusive. Or is its geometric, and relates to some kind of relation between events on distinct bases of a spin system.

When one speaks of an orthonormal bases or vectors that are mutually orthogonal , and are unit vectors that are mutually orthogonal, is this where ||u||=sqrt(x^2+y^2)=1 and ||v|=sqrt(x_i^2+y_i2)=1

u.v=x_2.v_2+ y_2.y_2=0;

does this denote, complementarity/non commutting, geometric orthonality, or logical orthogonality, that is events are disjoints in the logical sense (lie in the same basis or a commutting bases, and can be added) or are on distinct bases..

In other words

Is is there of way of using the inner product to determine whether the events are lie on non commutting and cannot be explicitely assume to add. I presume that the hilbert space inner product denoted logical orthogonality

x is amplitude moduli of spin up in direction of the vector u, with probability ||x||^2, and

y is the same but for spin down down in the same direction vector u with probability ||y||2

and x_i, y_i are the corresponding amplitude,moduli of spin up and spin down for the direction of the vector v,and probabilities,

||x_i||^2

||y_i||^2.

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.

Is the following function F:[0,1] to [0,1]

F strictly monotonic increasing F(1)=1,(i presume this unnecessary as its specified by the first two

(1)ie x+y=1 if and only iof F(x)+F(y)=1 F(x)+F(1-x)=1; F-1(x)+F-1(1-x)=1

(2)F homomorphic(2) x+y+z=1 if and only F(x)+F(y)+F(m)= 1

(3) x+y+z+m=1; if and only ifF(x)+F(y)+F(z)+F(M)=1

Give F(x)=x and F continuous (it appears to entail cauchys equation with F(1)=1 over ethe unit triangle due tot he common term). F(x+y)+F(z)=1, F(x)+F(y)+F(z)=1, etc, F(x+y)=1-F(z)=F(x)+F(y)

.

I

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

I was wondering if there are any set of n; n>=3continuous or somewhat smooth functions (certain polynomials), all of which have the same domain [0,1]

f^n(min,max):[0,1] \to[min,max]; where max>min, min, max\in [0,1] that have these properties

1.\forall v \in [0,1]\sum_n=1\toN(f^n(v)=1; for all values in the domain [0,1], the sum of the 'function values' =1at all points in this domain, and for all possible min and max values of the ranges of the n functions

2. The n functions have range continuously between [min,max], max>=min, both in [0,1]; and will all reach their maximum at some point, but only once (if possible).

3. These functions must reach their global maximum value only at the same point on the domain st,t the other distinct functions attain their global minimum value. These global minima for each fi, may occur at multiple points in the domain [0,1] .

4. There is, for each point in the domain, under which any distinct function fi\neqfj,( fj j\neqi) reaches a global maxima, at least one point (the same point) in the domain s.t that any fi attains its global minima. Correspondingly, each function fi has (or at least) n-1 points (in the domain) under which it attains its global minima, if each of the n functions has a singular point in the domain under which it attains its global maxima (the n-1 points corresponding to the distinct points in the domain under which the n-1 distinct functions, fj\neqfi attain their global maxima .

That is, for each, of the (n-1) global maximum values of the range of the other functions, f^i;kneqi, where there are n functions ; the value of the domain in [0,1] when one function reaches its maximum, is the same value of the domain wherein other functions at which they all reach their absolute minimum . W

5. The max values of the range of said functions, f^n= 1-\sum_{i;i\neqn}(min(ranf^i)). Ie the maximum value of the functions, f^i for all i\in{1,n} for n functions, is set by 1- sum (of the min function value/range of the other 'distinct' n-1 functions)

5. It must be such that whenever the minimum values are set so as to sum to one. ie min of any function= max of that same function for all such functions; that all the functions values become a flat line (ie, ie min=max, for such functions)

6.This should be possible for all possible combinations of min values of all 'n' f's (functiions), So each combination of n, non negative values in [0,1] inclusive that sum to one. So one should be able to set the min value of any given value of a function to any value in [0,1], so long as the conjoint sum, sums to one. Likewise presumable if any such function is set such that min=max, all of them are, and these will mins will add to one.

It must be be such that these functions should have to change, to make this work.So that for all min, max range values for the n functions and all elements in their domain (which is common to all and is [0,1]) these function values \forall(n, min function values)\forall(v in [0,1])\sumf^i(v) sum to one

6. Moreover and most importantly they must also allow for the non trivial case where the function minimums do not sum to one (ie are not all flat lines); but such that the sum of the values of the function of all such functions sum to one for all values in their common domain ,v=[0,1](which just is their domain, as their domains;f^i:[0,1]->[min,max] are the same).

Ie so that the functions will range between a maximum and minimum values (which are not equal) in a continuous and smooth fashion (no gaps, steps or spikes).

In this case it must be such that it possible for the the sum of the minimum values, to sum to some positive value smaller then one, although not greater than one, or smaller then zero; for all or 'some' possible combinations (obviously values may not be possible. although, not in virtue of the sum of the functions at some point not being one, but because the max has been set to be smaller then the min for some function; that is the sum of the mins is a particular combination greater then one; (ie 7 function min=0.16....., so that max of fi=1-1=0<0.166=min fi; that is if they sum to more then one;

7. I was wondering if there for any n>3,4,5,6,7,,,,four sets of such sets of 3,4,5,6,7 functions that will do this.

And then given this; it must also be such that the function can be modified so that a function only ever has a minimum of 0 if the maximum is 0; without having to set th sum of the mins of the function to 1.

And likewise such that the function only ever reaches a max of one if its min is one, unless under certain circumstancess; unless that is, it is possible to do this,for the same reason; without doing this having to set the sum of the mins to one. If indeed one does want some of the functions to range between. If possible, Although, What is most important, is that if any function has a min and max set to one, all of the rest of the functions values sit at zero for all v in [0,1].

It also must be such that if the maximum range value for any given function f1 is larger than that of another function f2 in the set, then f1's minimum value will be larger then that of f2's minimum range value; and conversely.

Are there sets of three, or rather for any N>3 sets of N

**sur-jective**uniformly continuous functions, for all N>3, where n denotes the number of function in the sets, such that each function in the set has the same domain [0,1] and the same range [0,1],such that these functions, in a given set, sum to one on all points in the domain [0,1]. ie \forallv\in[0,1] [\sum_{i=1}^{i=1n)fi(v)} =1 in [0,1] .Moro-ever, Are there arbitrarily many such functions for all n>=3.By non trivial, I mean, -non linear, (and presumably not quadratic) functions which just so happen to be that that their sum give value 1, in [0,1] or perhaps for any domain, and not by way of the algebraic sum cancelling out to a constant 1 as in the case of x, 1-x (ie error correcting) .Presumably due to the nature of the derivatives

That is, so that the functions, would sum to one, regardless of the domain [0,1] or least for if the domain [0,1] is held fixed, and the functions are weighted; without it being a mere artefact that the algebraic sum to cancels out to give a constant,1,for any x; that is error-correcting functions. That is three (or n>3) functions all of which have a maximum range value of one, miniimum range value of zero, and which sum to one for all points in the domain [0,1]; and which are surjective and uniformly continuous, ie for every value in the range ri\in [0,1], these functions have some (at least one) value, ci in the domain [0,1] such that f(ci)=ri such that takes on that .

Where these maximum values and min values for all fi coincide (the same element of the domain for which fi corresponds to 1, is such that other n-1/2 functions correspond to zero etc); where obviously these three/n max points such that fi(c)=1,(one, and only one for each function, fi,i1<=i<=n for n>=3, such functions) are distinct (correspond to distinct elements of the domain), although they correspond to the same element of the domain for for which each of the other 2/n-1 are at their minimum value (ie 0)

So for n=3 there is one one max point for each function, and at least , and presumably only, two/n-1 min points), s.t, when one functions fi(c) hits 1, fi(c)=1,the other 2/n-1 functions fj(c),jneqi take the value 0, fj(c)=0 for all j.And are there at least two such such sets for all sets of N>=3 such functions. So there are three distinct domain points in [0,1] corresponding to a n/3-tuple of function values,f1,f2,f3(c)= <1,0,0>,,f1,f2,f3(c1)=<0,1,0>f1,f2,f3(c2)=,<0,0,1>; corresponding the three distinct elements/points in the domain [0,1] , c\neqc1\neqc2, c,c1,c2,\in [0,1] . Where the first function is at its maximum value here 1, and the other two are their minimum, zero at c, the second is at its maximum at c1 (1), and first and third are a their mimimum (zero here) at c1 etc

And are continuous (no gaps, and uniformly continuous, ie no spikes).

As one could not make use of these such functions, if one wanted them to be weighted otherwise; if said functions have sums which either (A) as just cancel out to be constant, or two (are such that it their sums are do not cancel out to a constant, but just so happen to line up because the domain is [0,1]

Likwise, the functions a similar form; so that one does not want, one having two maximums whilst the other two for example have one maxima, and two minima. Perhaps Berstein polynomials could be so weighted, but I do not know; the linear forms cancel out but their weighted bezauir forms seem to be a little unstable from what I have read.

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?