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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7
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3answers
260 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, ...
-2
votes
0answers
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)&...
2
votes
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 ...
0
votes
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?
1
vote
1answer
38 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} \...
0
votes
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 $...
1
vote
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 ...
1
vote
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 ...
0
votes
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
votes
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}^...
1
vote
0answers
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 ...
0
votes
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?
0
votes
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. ...
1
vote
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}\...
2
votes
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
votes
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} ...
1
vote
0answers
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 $...
0
votes
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}})\,\...
0
votes
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 ...
3
votes
0answers
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 ...
0
votes
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 ...
1
vote
2answers
41 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 : $X_n$...
2
votes
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 }\...
1
vote
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 ...
0
votes
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 $5$ ...
0
votes
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 ...
0
votes
0answers
46 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 $i$ ...
-1
votes
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 $\mathbb E[X\ \mathbb{I}_{A}]$?
1
vote
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 ...
0
votes
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
votes
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 ...
0
votes
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 ...
1
vote
1answer
32 views

Closed Form E[exp(x'Ax)]

Is there a (general) closed form available for the following expression? $$\mathbb{E}\left[e^{x^{T}Ax}\right]$$ Where: $$x=\left\{ x_{1},x_{2},...,x_{N}\right\} \sim\mathcal{N}\left(0,\varSigma_{N}\...
1
vote
1answer
47 views

Probability of an Ornstein-Uhlenbeck process

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration, with $\tau < \infty$. The following definition is ...
0
votes
0answers
24 views

How to rotate an $n$-dimensional normal distribution, to maximize the likelihood of a sample

Suppose we have a normal distribution with a diagonal covariance matrix S and mean $0$, i.e. $N(0,S)$. I want to find a Rotation matrix $R$, to rotate this distribution to maximize the likelihood of a ...
1
vote
0answers
37 views

How to provide Mathematical Proof for number theory scheme?

I have a set S={1,2,...,N-1}. N=pq (where p and q are RSA prime numbers). Scenario is that User need to retrieve the Database blocks without revealing his block index to the Server i.e, Private ...
0
votes
0answers
38 views

Do $\limsup A_n$ and $\liminf A_n$ always have probability 1 or 0?

Is it right that, for a sequence of events $(A_n)_{n}$, $\limsup A_n$ and $\liminf A_n$ have probability $1$ or $0$?\ My idea for $\limsup$ is for example the following: $$P(\limsup A_n) = \mathbb{E}(...
-1
votes
1answer
58 views

Probability of getting n heads when n is very large [closed]

How to show that the probability of getting n heads when 2n times coin is toss is very small? Moreover, how to show that the probability of more than n heads is close to 0.5?
3
votes
1answer
33 views

Is the supremum of an almost surely continuous stochastic process measurable?

Let's take a stochastic process $(X_t)_{0\leq t \leq 1}$ and assume that the sample paths are almost surely continuous. Let us define $S \equiv \sup_{t \in [0,1]} X_t$. How can we show that $S$ is ...
0
votes
1answer
17 views

Rearranging infinite series (poisson distribution problem)

Calculate $$\sum_{n=1}^\infty 10,000(n-1)\frac{(3/2)^ne^{-3/2}}{n!}$$ The answer my solution guide gives is: $$\sum_{n=1}^\infty 10,000(n-1)\frac{(3/2)^ne^{-3/2}}{n!}$$ $$ =\sum_{n=0}^\infty 10,000(...
18
votes
3answers
185 views

How often was the most frequent coupon chosen?

In the coupon collector's problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how ...
3
votes
0answers
64 views

(Infinite hat)-guessing problem

$2$ men are playing a game: they are wearing countably infinitely many hats on their heads. The hats are either black or white with probability $\frac 12$. They see the other's man hats but cannot see ...
1
vote
0answers
17 views

Lipschitz constant/derivative of the stationary distribution of a Markov chain under perturbations in the transition kernel

I'm interested in the following question: Given a parameter $t\in \mathbb{R}$ and a column stochastic matrix $P(t)$ (i.e., $e^T P(t)=e^T$ and $P(t)_{ij}\ge 0$), calculate the Lipschitz constant of ...
3
votes
1answer
70 views

Is statistical physics background desirable for probability theory?

I am talking about higher probability viz. Brownian Motion, Ergodic Theory, Concentration, Percolation, Random Graphs, Random Matrix, etc. Going through books, I find that somehow or the other, many ...
1
vote
0answers
11 views

What is the “time change” of an adapted finite-variation stochastic process?

Let $(\Omega, \mathcal F,\mathbb P)$ be a probability space equipped with a filtration $\{\mathcal F_t:t\in\mathbb R_+\}$ satisfying the usual conditions of completeness and right-continuity. Suppose ...
4
votes
0answers
44 views

Stopping-time sigma-algebra and the case at infinity, definition question.

Assume you have a probability space $(\Omega,\mathcal{F},P)$, and you have a filtration $\{\mathcal{F}_t\}$ and a stopping time $\tau$. Then all the books I have seen define the stopping time sigma-...
2
votes
1answer
47 views

Use Poincare Recurrence to show existence of $n$

Suppose $A\subset \mathbb N$ such that $d(A)=\lim_{n\to\infty}\dfrac{|A\cap [1,n]|}{n}>0$. Then show there exists $n\in\mathbb N$ such that $\overline{d}(A\cap (A-n))>0$ where $\overline{d}(B)=\...
1
vote
1answer
28 views

Why $P\left(Y>X\right)=\sum\limits_n P\left(X=n\right) \cdot P\left(Y\geq n+1\right)$

Joint Distribution Chapter of P exam book—Discrete case. Problem 41.7 (p exam book by M. Finan) Part of the question's solution was already posted here. Michal's answer was: \begin{align} P\left(X=n\...
1
vote
1answer
30 views

expected value of game involving uniform variable and its square

I am trying to determine the expected value of the following game: Let $u$ be drawn from a uniform distribution on $[0,1]$. We write down $u$ on one side of a piece of paper and $u^2$ on the other ...
0
votes
1answer
22 views

Total Probability Theorem / Partition

In the Total Probability Theorem we assume that the sample space is partitioned into subsets. If we consider $B$ to be the sample space and $A_1$, $A_2$ to be the partition then the theorem says: $$\...