Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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30 views

How do I compute $P(X=Y)$? for independent random variables with with geometric distribution.

let $X$ and $Y$ be independent random variables with geometric distribution and parameter $p\in(0,1)$ How do I compute $P(X=Y)$? Any help would be greatly appreciated.
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2answers
26 views

Show $\lim\limits_{m\to\infty}P(n\leq m)=1$ for some function $n:\Omega\to\mathbb{N}$

Suppose that $(\Omega,\mathcal{F},P)$ is a probability triplet and $n:\Omega\to\mathbb{N}$ is some measurable function (in particular, $n(\omega)$ is finite for each $\omega\in\Omega$). I'm ...
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0answers
8 views

Counting minimal cut sets in nonamenable graphs

Suppose that $G$ is a fixed infinite, bounded degree, countable, connected nonamenable graph, meaning that its Cheeger constant is positive. Let $x\in G$ be a fixed vertex. I need to show (for ...
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2answers
53 views

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$?

If $P(A) < P(A \cup B)$, does that mean that $A\subsetneq (A\cup B)$? I thought that by monotonicity, which states that if $A \subseteq B$ then $P(A) \le P(B)$, then: If $P(A) < P(A \cup ...
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0answers
43 views

Question about $M/GI/ \infty $ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
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1answer
31 views

Not all finite exchangeable families can be infinitely extended

Give an example of a finite exchangeable family $(X_1, X_2, \ldots, X_n)$ which can not be extended to an infinite exchangeable family $X_1, X_2, \ldots$. Luckily, such a counterexample can be ...
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0answers
22 views

Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
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0answers
23 views

conditional expected value for multinomial distribution

Let $X = (X_1, \ldots, X_l)^T \sim Mult(n,p_1, \ldots, p_l)$. I've already calculated that $X_i \sim Bin(n,p_i), ~ i=1, \ldots l \\ P^{Z_i | X_i =k} \sim Mult(n-k, \frac{p_1}{1-p_i}, \ldots, ...
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0answers
20 views

Interpretation of conditional expectation as a random variable

I have a couple problems understanding the conditional expectation as a random variable. Consider the fair dice roll as a random variable $X$. Let $C$ be the event that the dice shows a one and ...
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99 views

Covariance inequality for infinitely many exchangeable random variables

Let $X_1, X_2, X_3,\ldots$ be exchangeable, square-integrable random variables. Show that $\mathbf{Cov}[X_1, X_2] \geq 0$. Solution: Assume that \begin{equation*} \mathbf{Cov}[X_1, X_2] ...
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1answer
87 views

Covariance inequality for $n$ exchangeable random variables

Let $n \in \mathbb{N}$, $n \geq 2$, assume that $X_1,\ldots, X_n$ are exchangeable, square integrable random variables with $\mathbf{E}\bigl[X^2_1\bigr] < \infty$. Prove that the following ...
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2answers
22 views

Density of conditional distribution

Let $X$ be a continuous random variable with density function $f(\cdot)$. Define $Y = 2X$, another continuous random variable. I would like to determine the conditional density of $f_{Y|X}(y|x)$. It ...
2
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1answer
28 views

Stochastic kernel as linear operator

Let $K$ be a stochastic kernel for a set $S$ equipped with a countably generated $\sigma$-Algebra $B(S)$, i.e. $K:S\times B(S)\rightarrow [0,1]$ such that $K(\cdot,A)$ is a measurable function for ...
2
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1answer
21 views

Natural structure over a set of measurable functions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $U$ be the set of all measurable functions over $(\Omega, \mathcal{F}, \mathbb{P})$ - i.e. the elements of $U$ are all measurable ...
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0answers
16 views

Optimal $A\in \Sigma$ that maximizes an objective

Let $([0,1],\Sigma, \lambda)$ be a probability space. For any given $B\in \Sigma$, $K\in [0,1]$ and $f\in L^2(\lambda)$ with $f(x)\in[0,1]$ for all $x $, $$\max_{A\in \Sigma}\int_A f(x) d\lambda(x)- ...
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0answers
25 views

Problem on “Distribution” of conditional expectation w.r.t a sigma field [duplicate]

Let $Y$ be a random variable s.t. $E|Y|<\infty$. Let $E[Y|\mathcal{G}]$ and $Y$ have the same distribution. I need to prove that $E[Y|\mathcal{G}]=Y$ a.s. How does one use the fact that a random ...
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2answers
34 views

R.V. X and Y have joint density $f(x, y) = e^{(-x-y)}$, find $P(X+Y\leq 1)$

What I did was let $Z = X + Y$ and then did a double integral, where $x$ is integrated from $0$ to $1$ and $y$ is integrated from $0$ to $Z - X$, but that gave me $$-e^{-z} - e^{-1} +1$$ which is not ...
3
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1answer
46 views

Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of ...
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1answer
45 views

The support of a the sum of random independent variables

The support of a continuous random variable is the set of the outcomes such that $f(x)>0$. If $X$ has support $[a,b]$ and $Y$ has support $[c,d]$ and they are independent, what is the support of ...
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1answer
32 views

linear least square estimation with random sum

Let $N$ be a geometric r.v. with mean $1/p$; let $A1,A2,… $be a sequence of i.i.d. random variables, all independent of $N$, with mean $1$ and variance $1$; let $B1,B2,… $be another sequence of i.i.d. ...
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1answer
18 views

Differentiability of CDF at 0

This might seem to be a very trivial question but anyway here we go: I'm currently reading the paper "On the Value of a Random Minimum Spanning Tree Problem" by Frieze (1984) and I'm stuck on the ...
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1answer
49 views

Proof of martingale representation theorem monotone class argument

Martingale representation theorem for reference: Theorem: (Martingale Representation) Let $M$ be a square integrable Brownian martingale with $M_0 = 0$.Then there exists a process $X$ which is ...
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1answer
25 views

expectation of matrices with random components

Let's say I have a matrix where some of the components are random variables. From Wikipedia, the expectation of the matrix is simply the expectation of the individual components. But, beyond that, ...
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0answers
27 views

Empirical distribution generates exchangeable $\sigma$-algebra

I have a problem understanding the following statement from Klenke, p. 234: If we write $\Xi_n(\omega) := \xi_n \bigl(X(\omega)\bigr) = \frac{1}{n} \sum^n_{i=1} \delta_{X_i(\omega)}$ for the ...
1
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1answer
27 views

Notation for a statistic, or function of a random variable

A statistic is a function of random variables, so it is also a random variable. Suppose we have a collection $X = (X_1, X_2, \dots, X_n)$, where $X:\Omega \to \mathcal{X}^n$. There are two common ...
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0answers
31 views

For any given value x, are there uncountably many (countably infinite) binary sequences (ones and zeroes) whose limiting relative frequency is x

I have the following question, and given few proofs (provided by friends, professors, and my myself) which seem to work, I suspect the answer is yes: But I am still not completely sure. The question ...
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0answers
36 views

$\sigma$-algebra of events invariant under permutations

Let $X = (X_n)_{n\in\mathbb{N}}$ be a stochastic process with values in $E$. For $n \in \mathbb{N}$, define $$\mathcal{E}'_n := \sigma\bigl(F : F : E^\mathbb{N} \rightarrow \mathbb{R} \text{ ...
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0answers
45 views

2d random walk on the nonnegative quadrant using martingale techniques

I know the basics of (discrete time) martingales, and I'd appreciate any help and suggestions on how to prove the following using martingale techniques. Let $Z_n$, $n\ge 0$ be a random walk on the ...
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0answers
19 views

Convergence of vectors

Recently I've read a paper and there is one moment I cannot fully realise on my own. It states as follows. There is a vector of estimates $\hat{\mathbf{X}} = (\hat{X}_1, \dots, \hat{X}_N)$ (N is ...
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0answers
26 views

Log of characteristic function

I am looking at a proof related to CLT and I have difficulties understanding a condition in the theorem. Basically, the theorem is trying to show that $\frac{d^m}{d u}ln[\varphi(u)]\leq k \beta_n $ ...
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1answer
20 views

Calculating the number of arrangements of items into compartments?

I'm dealing with intro probability theory here, and am a bit perplexed by the logic around arranging items in boxes. When arranging a small set of objects in a series of boxes, I am not seeing why the ...
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1answer
36 views

How to rewrite this probability formula?

In probability, I know that $$P(C=1|W=1)=$$ $$\dfrac{P(C=1,W=1)}{P(W=1)}$$ But what if I have variations like: $$P(C=1|W=1,R=1) and P(C=1,R=1|W=1) and P(C=1 or R=1|W=1) and P(C=1|W=1 or R=1)$$ ...
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2answers
24 views

Use of permutations and combinations for a conditional seating arrangement

I've got a problem where there are 6 seats in a row, with 6 people to be seated. I need to figure out all possible seating arrangements, given that two people are married and are to be seated next to ...
0
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1answer
13 views

Does the natural (asymptotic) density of a set A change if a subset of A with natural density zero is subtracted from A?

I know that given two subsets of the Naturals A and B, if the natural density of A equals some non-zero real number a, and the natural density of B is zero, then the natural density of the symmetric ...
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0answers
24 views

Simplification of an expression sums and products

I am trying to simply the following expression: $ \sum_{\substack{\overline{t}_i , \overline{t}_j \in \{H,L\} \\ \forall j \in N_i}}^{} \sum_{j \in N_i}^{} \alpha_{\overline{t}_i , \overline{t}_{j}} ...
2
votes
1answer
21 views

Using Central Limit Theorem to show that random walk exits a interval a.s. in finite time.

Let $X_0 = x \in \mathbb{Z}$ and $X_1, X_2, \dots$ are i.i.d. random variables with values in $\{-1,0,1\}$ all with positive probability and $E(X_1) = 0$. Let $\sigma^2 = E(X_1^2)$. Let $S_n = ...
1
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2answers
27 views

Variable transformation $Y= \sin (X)$ where $X$ is uniform on $[0, \pi]$

Given $X$ is uniform on $[0, \pi]$. Let $Y= \sin (X)$ then $y \in [0,1].$ What is the distribution of Y? If intuition serves me correct I would think that Y is uniform on $[0, 1]$. But when I start ...
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0answers
34 views

how to determine transient and recurrent state from transition matrix

I wonder how can I determine the transient and recurrent state from transition matrix ? I mean if I have 10 states It would be very hard to draw diagram for them so how to analyse the matrix? For ...
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0answers
17 views

Does every measurable function create a partition of its domain?

Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be two measurable spaces. Does every measurable function $f\colon S \to T$ create a partition of $S$? If $T$ is finite $T=\{C_1,\dotsc,C_n\}$ then this is ...
0
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1answer
22 views

CDF of two variable

I would like to calculate the CDF of sum of two random variable in a unit square I realize that everywhere says if X+Y=z and then if z is between 0 and 1 then probability is equal to something and if ...
0
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1answer
67 views

Infinite coin flipping

Suppose that two players are flipping the coin (which is assumed to be fair). They continue to flip up to the moment when either of two sequences occurs: HH or TTT. In the first event player 1 wins ...
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0answers
27 views

Two-parameters Wiener process

Two-parameters Wiener process $W(r, u), r \in [0, 1], u \in [0,1]$ is a stochastic process with a covariate kernel $\mathbb{E}\left[W(r_1, u_1) W(r_2, u_2)\right] = \min(r_1, r_2) \min(u_1, u_2)$. ...
1
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1answer
35 views

How many infinite subsets of the Naturals have natural density (asymptotic density) zero?

Are there countably or uncountably many? I know that the set of all primes has density zero. Is there an obvious way of using that result to construct an uncountable family of such sets?
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0answers
42 views

combinatorics,defects on disks

$k$ defects are randomly distributed amongst $n$ computer disks produced by a company AND any number of defects may be found on a disk and each defect is independent of the other defects Let $p(k,n)$ ...
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0answers
16 views

Uncountable set of estimates on conditional expectations.

Let $X, Y$ and $Z$ be real-valued integrable random variables. If we assume that for all $r \in \mathbb{R}$ $$\mathbb{E}\left[ X \middle| Z = r\right] > \mathbb{E}\left[ Y \middle|Z =r \right],$$ ...
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0answers
22 views

T invariant probability measure on a compact space $X$

Let $T:X\to X$ be a continuous map defined on the compact space $X$. I read that if $X$ is a metric space then the set of T-invariant probability measure is not empty. I want to know if this result ...
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0answers
17 views

Unfair coin tossing game,target,optimal fixed investment,

Suppose the player has capital 1\$. He chooses a number $f\in[0,1]$.He tosses an unfair coin repeatedly, which wins for him, with probability $p$, a gain $q\times f \times$ current capital \$,where ...
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0answers
51 views

Exponential moments and integrability

Given a probability distribution $P(x)$ for which the moments $\int x^rdP(x)$ and the exponential moment $\int e^{Rx}dP(x)$ exist for some positive integers $r$ and $R$, than is it true that ...
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0answers
22 views

inhomogeneous poisson with exponential lambda function

Suppose that a random variable X is distributed according to a Poisson distribution with parameter $\lambda$. The parameter $\lambda$ itself is a random variable, exponentially distributed with ...
1
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0answers
38 views

Lindeberg-Feller CLT follows from Martingale CLT?

I've been studying about central limit theorems, in particular the Lindeberg-Feller CLT and other extensions. In most textbooks and sources online the martingale central limit theorem comes as an ...