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 [tag:probability-...

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property of universally measurable function

Let $\mathcal B_d$ be the Borel $\sigma$-algebra on $\mathbb R^d$ and define $\mathcal B_d^*= \bigcap_{\rho} \overline{\mathcal B_d}^{\rho}$, where $\overline{\mathcal B_d}^{\rho}$ is the completion ...
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1answer
43 views

Is there a way to write conditional expectation as an integral?

Let $E[X|G]$ be a random variable that is G-measurable and satisfies the partial averaging property, then we know that $E[X|G]$ is a conditional expectation. This is the definition I saw from Shreve's ...
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26 views

Transformation of random variables exercise

I want to know if my solution to the following exercise is correct: Let $X$ be a gamma distributed random variable with parameter 2, meaning with distribution $$P_X(\mathrm{d}x)=\mathbb{1}_{\{x>...
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40 views

On the distribution and the moments of $\max\{1/\sqrt{U_1},…,1/\sqrt{U_n}\}$, where $(U_k)$ is i.i.d. uniform on $(0,1)$

Let $U_1,U_2,...$ denote an i.i.d. sequence of random variables with the uniform distribution on $[0,1]$. For every integer $n\geq1$, we set $M_n = \max\{1/\sqrt{U_1},...,1/\sqrt{U_n}\}$. a) Compute ...
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2answers
54 views

Show that the sum of a sequence of random variables converges almost surely

Let $(X_n)_{n\in\mathbb N}$ be a sequence of non-negative iid random variables with $\mathbb E[X] < \infty$. How could one go about showing that $\sum^{\infty}_{k=0} e^{X_k} c^k < \infty$ ...
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1answer
21 views

Property of renewal point processes

For a renewal process where $f(t)$ is the number of arrivals in time $t$ and $S_k$ is the $k^{th}$ time of arrival, how can we show: $$f(\alpha S_k)/k \xrightarrow{\text{a.s.}}\alpha $$ as $k \...
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0answers
47 views

Probability of sum of 10 dice throws [duplicate]

If a die is rolled 10 times. What is the probability that the sum of the results is less than or equal to 20? I was trying to solve this using something like $P(X_1 + X_2 + ....+X_{10} \le 20)$ but ...
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1answer
42 views

Does absorbing Markov chain have steady state distributions?

If I am not mistaken, the steady state distribution is independent of initial state distribution, and regular Markov chains satisfies this definition. On the other hand, since the row of each ...
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1answer
36 views

Suppose a sequence's subsequences have at least one subsubsequence that converges almost surely to $X$. Prove convergence in probability

Probability with Martingales What I tried: 'only if' Suppose a sequence converges in probability to $X$. By $d$ there exists a subsequence that converges almost surelyto $X$. Then by $a$, ...
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1answer
55 views

Does $\mathbb E[X] < \infty$ imply that $\mathbb E[e^X] < \infty$? [closed]

Let $X$ be a non-negative random variable with finite expectation. Is this sufficient to say that $\mathbb E[e^X] < \infty$? Jensen's equality clearly shows that if $\mathbb E[X] = \infty$ then ...
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11 views

what are the mean vector and covariance matrix of the multivariate truncated normal distribution?

Let $\mathbf{X}=(X_1,X_2, \ldots, X_p)^\prime$ has multivariate normal distribution with mean vector $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$, i.e. $\mathbf{X}\sim N_p(\mathbf{\mu}, \...
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0answers
21 views

Indicator function for a vertex-induced random subgraph of $G$?

I am trying to find polynomial, indicator function or sometimes called structure function to express whether a vertex-induced random subgraph $H$ of $G$ is connected or not. The polynomial $\phi(G')$ ...
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1answer
39 views

Stochastics exam Exercise

The professor uploaded an exam to practice, but unfortunately I have no solutions. Let U be a unifomly distributed random variable on $[0,1]$. 1) Let $X=-ln(U)$. Show that $X$ is distributed ...
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1answer
39 views

Conditional expectation and independence on $\sigma$-algebras and events

In many statistics papers, proofs might proceed as follows: Under the event $A$, the random variables $X$ and $Y$ are independent. (Often this means that on $A^C$, they might be dependent). Then some ...
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0answers
8 views

How to prove pairwise independence of a family of hash functions?

I want to prove pairwise independence of a family of hash functions, but I don't know where to start. Given the family of hash functions: H with h(x) = a * x + b (mod M). ( Say h: U -> V, then: M ...
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1answer
32 views

Double Integrals and Joint Probability

An insurance policy is written to cover a loss X where X has a density function $$f(x) = \frac{3}{8}x^2; \qquad 0 \le x \le 2$$ The time (in hours) to process a claim of size x, where $0 \le x \le ...
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1answer
46 views

Expectation of $|X-Y|$ when a coin is thrown six times

If a fair coin is thrown six times. Let $X =$ number of heads and $Y = 6-X =$ number of tails. What is $E|X-Y|?$ I was able to come up with this table, but I am not sure if this is correct or not and ...
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34 views

Loaded Dice Conditional Probability

If I have two dice, one regular and one loaded. The loaded die has the probability 1/2 of landing a six and rest of the numbers are equally probable. If you select a die randomly and throw it and it ...
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3answers
262 views

What does it mean to integrate a Brownian motion with respect to time?

I am reading about stochastic process, but could not make sense if one equation I encountered. Can anyone help me understand it? The equation states that suppose R(s) is an interest rate process, ...
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26 views

Phase transitions and Infinite Cluster in Finite Percolation?

The percolation probability is defined with respect to infinite cluster such that $\theta(p)=P(|C|=\infty)$. The critical value $p_c=p_c(d)$ of $p$ such that $\theta(p)=0$ if $p<p_c$ and $\theta(p)&...
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2answers
43 views

Number of Spaghetti loops

From Peter Winkler's book: the 100 ends of 50 strands of spaghetti are paired at random and tied togethed. How many pasta loops should you expect from this process on average? I took ages because ...
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1answer
17 views

Joint distribution of $n$ Bernoulli variables equal to binomial distribution, how? [closed]

Is the joint distribution of $n$ Bernoulli variables equal to binomial distribution? I am confused by this questions and I would like to understand this. What about if Bernoulli variables dependent?
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1answer
39 views

Law of large numbers along moving window

The law of large numbers says that if $X_1, X_2, \ldots$ is an i.i.d. sequence, with $\mathbb{E}|X_1| < \infty$, then $$ \frac{1}{n}(X_1 + \cdots + X_n) \stackrel{a.s.}{\longrightarrow} \...
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0answers
19 views

Quantizer Functions

Let $Y \sim P_Y$ with variance $P^{\alpha_1}$ $P>1$. Assume $n \sim P_n$ with variance $P^{\alpha_2}$ for any $\alpha_2 \le \alpha_1$. Let $\mathcal{Y}$ be the set over which the random variables $...
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2answers
72 views

What is the diagonal principle?

I'm guessing I'm supposed to show that there exists a subsequence that converges in probability quickly (hypothesis of $(c)$). What is the diagonal principle? Is that related to Cantor's ...
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2answers
154 views

Chance of receiving all elements of a set

I have a set of $n$ different elements. I will select $i$ times a subset $S_j$ of $n/k$ elements randomly. Each element can only occur once in each $S_j$, but can be part of multiple different subsets ...
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0answers
15 views

Absorbing State vs Closed Communicating Class

According to Wikipedia, A set of states C is a communicating class if every pair of states in C communicates with each other. A communicating class is closed if the probability of leaving the class is ...
2
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1answer
23 views

Are derivatives of a characteristic function bounded?

Let $X$ be a real valued random variable with cdf $F(x)$ and characteristic function $\varphi(t)$, and suppose that $E[|X|^n]<\infty$ for some $n$. Then we know $$\varphi^{(k)}(t)=i^k\int_{-\infty}^...
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37 views

Is it an absorbing state if it does not communicate with other states?

$$\begin{matrix} 1 & 0 & 0\\ 0 & 0.5 & 0.5\\ 0 & 0.5 & 0.5 \end{matrix}$$ Given that this is a right transition matrix, would you call the state in the first row, say A, an ...
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1answer
22 views

convergence in probability of division and their expected values

Let $\frac{X_n}{Y_n} \rightarrow 1$ in probability. Then does $\frac{\mathbb{E}[X_n]}{\mathbb{E}[Y_n]} \rightarrow 1$? If not, what are the conditions required for this to hold?
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1answer
10 views

Expectation of Bivariate Distributions

I know very little about probability and I was searching for the expected value of bivariate distributions, but all I could find was the expected value of a real-valued function of the distribution. ...
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1answer
99 views

Measurable Functions with Common Sigma Sub-Algebras

Let $X:\mathbb{R}\rightarrow\mathbb{R}$ be a non-constant function, measurable with respect to the Borel-Algebra $\mathcal{B}$ and $\sigma(X)$ the sigma-algebra generated by $X$. Let $\mathcal{A}\...
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0answers
24 views

Marginal mean of two dimensional Brownian motion.

Let $(B^1,B^2)$ be a two-dimensional Brownian motion. Let $t>s$. Is it true that $$ E[B^1(t) \lvert B^2(t),B^2(s) ] = E[B^1(t) \lvert B^2(t),B^2(t)-B^2(s) ] = E[B^1(t) \lvert B^2(t)] $$ since $(x,...
2
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1answer
38 views

Determine the values of $r$ for which $\lim_{N\rightarrow \infty} \frac{\Sigma_{n=1}^{N}X_n}{\Sigma_{n=1}^{N}n^r}=1$

Let $X_n,n \geq 1$, be independent random variables s.t. each $X_n$ has Poisson distribution with mean $n^r$ for some real number $r$. Determine the values of $r$ for which $\lim_{N\rightarrow \infty} ...
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20 views

Does a linear operator on probability measures determine a Markov kernel?

Let $(\Omega, \mathcal{F})$ be a measurable space. Let $M$ be a linear operator on the space of probability measures on $(\Omega, \mathcal{F})$, i.e. for $\alpha \in [0,1]$ and probability measures $...
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1answer
25 views

Expected Value of Order Statistics with non IDD R.V.

Let $$\begin{align} F_X(x) =&~ (0.03x^2-0.002x^3)\,\big[ 0 \le x \le 10\big]+\big[x>10\big] \\F_Y(y) =&~ y/10 \; \big[0\le y\le10\big]+\big[y>10\big] \\F_Z(z) =&~ (1-e^{{-z}/{5}})\,\...
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1answer
65 views

Will this integral be progressively measurable?

Assume you have a function: $F(t,x,\omega)$: $[0,T]\times E\times \Omega \rightarrow \mathbb{R}$, which is predictable (predictable is explained below). Each of the three spaces can be viewed as 3 ...
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63 views

Family of partitions, s.t. the quadratic variation of a BM diverges a.s.

This question is about a specific step in the solution of exercise 1.13 a) of the book "Brownian Motion" by Peres and Mörters (https://www.stat.berkeley.edu/~peres/bmbook.pdf). The exercise is on page ...
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0answers
26 views

compound probability and conditional expectation

I'm stuck on a formula which looks obvious but that I fail to prove: If $Z$ is a real-valued random variable with distribution $\mu$, $T_z$ is a random time for each $z\in \mathbb{R}$ and $B$ is a ...
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2answers
43 views

Almost sure convergence implies convegence in distribution - proof using monotone convergence

I'm trying to understand the following proof of the statement : "Almost sure convergence implies convegence in distribution" The definition of convergence in distribution is given as follows : $X_n$...
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1answer
34 views

Difference of two random variables and convergence

I have a question concerning this task: Let $X$ be a random variable and $X_n=X+Y_n$ where $$E[Y_n]=\frac{1}{n}\quad\text{ and }\quad\operatorname{Var}(Y_n)=\frac{\sigma^2}{n}\quad \text{where }\...
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1answer
46 views

Binary Stochastic Programming with Independent or Positively Correlated Co-efficients

A manufacturer can select a maximum of $N$ stores to fulfill orders from a total of $M$ stores who are looking for inventory, $N\le M$. The case when $N\geq M$ is trivially solved when all stores ...
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1answer
60 views

die game where we roll until we get a 5 or a 6

We roll a die until we get a $5$ and a $6$ for the first time, not necessarily consecutively and not necessarily in that order. We need to pay $x$ dollars before each die throw, and once both a $5$ ...
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1answer
32 views

Why is the characteristic function of a probability distribution function uniformly continuous? [duplicate]

Why is the characteristic function of a probability distribution function uniformly continuous? This is from page151 of Chung's A Course in Probability Theory. Specifically, why is the last ...
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48 views

Birth-death process and transience.

I am unable to tackle part c) and d) can anyone help/ sugesstions? A Markov chain with state space ${0,1,2,...}$ is called a “birth-and-death chain” if the only non-zero transitions from state $i$ ...
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1answer
19 views

Expectation of a random variable and an indicator function [closed]

Suppose you have a random variable $X$, and an event $A$. How do you evaluate the expectation $\mathbb E[X\ \mathbb{I}_{A}]$?
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3answers
40 views

Given MGF of X, find MGF of $ Y=X_1\dot\ X_2 \dot\ X_3$

Let $X_1$, $X_2$, $X_3$ be a random sample from a discrete distribution with probability funciton $p(0)= 1/3$ $p(1) = 2/3$ Calculate moment generating function, $M(t)$, of $Y=$$X_1$$X_2$$X_3$ My ...
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0answers
18 views

Positive moments of independent variables are also independent

Suppose we have $X$ and $Y$ which are random variables and they are also independent and we also have $i, j \in \mathbb{N}_{+}$. Is it true that $X^{i}$ and $Y^{j}$ are independent? Actually I need ...
2
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1answer
43 views

Uniformly distributed differences

Is there a collection of random variables $X_1,X_2,\ldots,X_n$ such that $Y_1=X_1-X_2,~Y_2=X_2-X_3,\ldots,~Y_n=X_n-X_1$, are independently uniformly distributed on $[-1,1]$. How $X$'s should be ...
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2answers
78 views

Equivalent random variables and sigma algebras

Consider two random variables $X$ and $Y$ defined on the same probability space $(\Omega,\sigma,P)$. We know that they are equivalent in the sense that $P(\{X \ne Y\})=0$. Let $A_X$ and $A_Y$ be the ...