Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,022 questions
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Expected value of a logarithm of a Levy process
I have a strictly positive Levy process $(L_t)$ with no Brownian part, drift $\gamma$ and jump measure $\nu$. Is it possible to calculate the expected value of the logarithm of this process, so $\...
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227
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Branching process question
(Cross-posted to math stackexchange question 130154)
I am trying to analyze the following branching process. We start with a root (level 0) node. Each surviving node has two children, each of which ...
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187
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Sampling when given a set of marginal distributions
There is an unknown joint multivariate distribution P(A_1, A_2, A_3, ..., A_n) (in my scenario, it's a n-dimensional contingency table), which we need to sample from.
Given an arbitrary set of ...
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127
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A problem about partial sum of random number composition
Consider the strong random number composition,
$x_1 + x_2 + \cdots + x_n = m$, with $x_i > 0$ and all possible compositions have the same probability.
Let random variable $S_i = \sum_{j=1}^i x_j$...
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1
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207
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Copulas and marginals thereof
Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following ...
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165
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Joint Probability that contains a variable and its Fourier Transform
Given the vector $\mathbf{d}$, where $\mathbf{d}\in\mathbb{C}^{N\times 1}$, we have two variables
$X = \mid\mathrm{F}[d]\mid^2,\quad\quad X\ge 0$
$Y = a+b (\mathrm{d}^H\mathrm{d})\quad Y\ge 0$
...
- 447
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493
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Simulating conditional expectations
There is a multidimensional process X defined via its SDE (we can assume that its a diffusion type process), and lets define another process by $g_t = E[G(X_T)|X_t]$ for $t\leq T$.
I would like to ...
- 667
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337
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What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?
This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
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479
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Passage Time Distributions for Poisson processes.
Let $(X_t)_{t \geq 0}$ be a standard Poisson process with intensity $\mu$. Let $\tau_b = \inf ( t>0 : X_t= at + b)$, where $a>0$ and $b<0$, and let $\sigma = \inf (t>0 : X_t \geq at)$. ...
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184
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Integration of discounted normal distribution
Hi
I want to find expectation of integration of normal distribution $\varphi(t)\sim N(0,\sigma\sqrt t)$
but i also want to discount it continuously with parameter $\alpha$.I mean i need to ...
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458
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Bounding mutual information given bounds on pointwise mutual information
Suppose I have two sets $X$ and $Y$ and a joint probability distribution over these sets $p(x,y)$. Let $p(x)$ and $p(y)$ denote the marginal distributions over $X$ and $Y$ respectively.
The mutual ...
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163
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"Reverse" stochastic dominance
Let $\mu$ and $\mu'$ be probability measures on $\lbrace0,1\rbrace^\Lambda,\:\: \Lambda:= {\lbrace 0,1,\ldots,n\rbrace}$. Assume that
$\mu(X_i=1|X = \zeta \text{ on } \Lambda \setminus \lbrace i\...
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383
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"X \in \cdot" in Probability Measure [closed]
My question is quite simple, but I was unable to find an answer by googling, since you can't exactly google syntax. What does the $\in \cdot$ mean in:
$$\lim_{n\to\inf}||P(S_n\in\cdot)-P(S_n+k\in\cdot)...
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293
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Open Jackson network with deterministic arrivals.
Dear Friends,
Is there any known Jackson-like theorem for an open Jackson network with deterministic arrivals?
Thanks,
Michael.
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319
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Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...
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578
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One-Variable Optimization Problem
$W_{opt}=\arg \{\max(\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha )\}$
subject to $\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$
We should find analytically the optimal $...
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343
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Can KL divergence go to 0, but $E[\log(p/q)^2]$ diverge in certain cases?
Let $p(x)$ be a fixed distribution over a discrete space.
Let $A, C > 0$ be constants.
Let $\epsilon > 0$. Can we find an example of a distribution
$q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\...
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138
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Why do I not use post hoc tests with a 2 x 2 factorial?
I know this is an obvious answer. I am probably over thinking what I'm doing, but I cannot recall. Does it have to do with not having enough variables to compare the various means?
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164
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Transforming to uniform numbers
Hi
I have a time series of probabilites, vector X
I need to convert the probabilites to uniform numbers.
As I understand it if I put the series into the cdf the output is thus uniform.
The problem ...
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574
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What's the expected number of iterations for this process?
Each step of the process consists of choosing a random integer between 1 and the last number chosen this way. On average, how long does it take to obtain "1" as a result of this process for any given ...
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377
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Robust entropy-like measure for analyzing uncertainity
I'm looking for a measure to analysis the uncertainty observed in a set of variables (with multivariate Gaussian distribution). So, I've tried conventional Shanon entropy (differential entropy) which ...
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6
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2k
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Chances to win an election
Let's say that tomorrow national president election is held. A poll asks 1100 persons which of the two candidates, A or B, will he or she will vote. 750 say will vote A, and 250 say will vote B. What ...
- 330
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2
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409
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$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]
$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is ...
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3
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3k
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Monte Carlo method and possible applications to computer poker?
I want to do something about ”games of incomplete information“，like "Computer poker program".I know,Albert university(in canada) have do a lot of things to that field,they write a program called: "...
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3
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215
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Proving the uniform distribution maximizes the expected value of the product of a random draw of $m$ elements from discrete distribution
Say I have a discrete probability distribution $p_i$, so $0 \le p_i \le 1$ and $\sum_i{p_i}=1$. We sample $m > 1$ draws $D$ from this distribution proportional to $p_i$ with replacement, and ...
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2
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251
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$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
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217
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Expected number of balls left out when choosing $n$ times from $n$ balls
I am given $n$ balls. For $n$ times, I pick one of them with uniform probability and put it back after picking it.
Let $U$ be the number of balls I have never picked, so $U\in \{0,\ldots,n-1\}$. We ...
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869
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How can I calculate the expected ranking of a competitor from the probabilities of each competitor reaching first place?
Say I have several competitors contending over some prize. I know the probabilities that any particular one of them will win the prize. It is assumed that the competitors all want to achieve the ...
-1
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2
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440
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$\langle X\rangle_t = t$
Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^d$ and $X_t = |B_t|$. What is the easiest way to see that$$\langle X\rangle_t = t?$$I need this result for a simulation I am running...
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407
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Conditional expectation: commuting integration and supremum
Let $X$ and $A$ be compact Polish spaces endowed with Borel $\sigma$-algebras. Let $\mathcal{A} = X\times \mathcal{B}(A)$ be the $\sigma$-algebra consisting of cylinders whose projections on $A$ are ...
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200
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Is the unordered sum of measurable functions measurable?
Let $E$ be a normed $\mathbb R$-vector space and $I$ be a nonempty set. Remember that $(x_i)_{i\in I}\subseteq E$ is called summable if there is a $x\in E$ such that for all $\varepsilon>0$, there ...
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1
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259
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Which type of convergence for this sequence of random variables? [closed]
Suppose that $X_1,X_2...$ is a sequence of non-negative real random variables. I have that $\mathbb{E}(X_i^2) \to 0$ as $i \to +\infty$, therefore my sequence converges at least in distribution to the ...
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1
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163
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Is it true that if a random vector has independent coordinates each bounded by $1$ then $P[ \|X\| \leq \epsilon\sqrt{n}] \leq (C\epsilon)^{n}$?
I'm studying Vershynin's well-written book on "High Dimensional Probability" and the third chapter on concentration of random vectors.
Exercise 3.1.7 from the book is the following.
Let $X =...
- 401
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1
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223
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Why do we define independence for zero-probability events? [closed]
I am learning about probability and the definition of pairwise independence is given as $P(AB) = P(A)P(B)$. My textbook motivates this definition as one to capture the intuition where the knowledge of ...
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1
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175
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A strange probability inequality
I need help to understand the following :
For any non-negative random variable $\zeta$: $\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)\leq\mathbb{E}(\zeta)\leq 1+\sum_{k=1}^{\infty}\mathbb{P}(\zeta\geq k)...
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1
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88
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Can we say that: $ \sum_{n\geq 1}{\frac{1}{n}(f_n(\omega)-g_n(\omega))}<\infty\qquad a.e $
Let $(\Omega,\mathcal{A},\mu)$ be a finite mesure space, and $\{f_n\}$ and $\{g_n\}$ two $L^1$-bounded sequences, such that :
$$
\sum_{n\geq 1}{\frac{1}{n}(F_n(f_n)(\omega)-g_n(\omega))}<\infty\...
-1
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1
answer
332
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Expectation where linearity does not hold
We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has the same distribution. Bad news is EX and EY may not exist but E(W+Z) is zero. Could we conclude ...
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1
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338
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about Function of Random variables [closed]
Hello,
I am studying random variables.
Question is this:
if rv X & a function g is known, what is the pdf of random variable Y = g(x)?
in the textbook answer is explained as follows.
P[y ≤ Y ≤...
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1
answer
168
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Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
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1
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148
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Weighted sum of zero-mean random variables
Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$.
Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...
- 367
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votes
1
answer
304
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The category Prob of finite measure spaces does not admit all products [closed]
I am currently working in a category called Prob which has objects which are finite measure spaces and morphisms which are measure preserving maps between the spaces. A map $f:X\to Y$ is measure ...
- 91
-1
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1
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173
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The distribution of the sum of a non-zero vector with random signs
Given a non-zero high-dimensional vector, $v\in (\mathbb{R} \setminus \{0\}) ^ d$, and a random sign vector $s \in \{-1,1\}^d$ (i.e., each entry is a rademacher random variable).
Empirically, I find ...
- 121
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1
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129
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Is it possible to regress an arbitrary function from a span?
Suppose we have three sets $A, B,C$ and a span $S := A \leftarrow C \rightarrow B$. There is a special case when the data of the span $S$ exactly specifies a function $f: A \rightarrow B$. In ...
- 1,313
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1
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122
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Approximation of function in general measure space
Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
- 411
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2
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614
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Bounded difference functions and sub-Gaussian random variables
We have the following standard theorem : Let $X$ be some set and $g : X^n \rightarrow \mathbb{R}$ be a measurable function such that it satisfies the ``bounded difference property" i.e $\exists$ $\{...
- 2,246
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1
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197
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Surely recurrent random walks and the law of the iterated logarithm [closed]
Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with
$$
P(X_i=1)=P(X_i=-1)=1/2,
$$
and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As ...
- 965
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votes
1
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558
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For i.i.d X and Y , if X + Y and X - Y are independent, show X is normally distributed [closed]
The question goes as follows:
If $X$ and $Y$ are independent and identically distributed, their density function $f(x)$ is strictly positive and second-order continuously differentiable. If $X+Y$ and $...
- 9
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2
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462
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How to deal with this Chicken-And-Egg problem ?
Let's imagine designing an odds pattern for a game, in which players bet for win or lose.
Suppose the probablity of winning is $p$, thus the probablity of losing is $1-p$.
Now imagine $n_1$ people ...
- 99
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1
answer
1k
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Approximating expectation [closed]
if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points:
E[f]=(1/N)(summation of f(x) over these N points).
...
-1
votes
1
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247
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Concentration results for non-standard Gaussian random vectors.
Given a $c$-Lipschitz function $f(X):\mathbb{R}^d \rightarrow \mathbb{R}$, and given that $X \in \mathbb{R}^d$ is a Gausssian random vector centered at $\mathbb{w} \in \mathbb{R}^d$ (not at zero) ...
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