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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4
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1answer
31 views

References for information theoretic statistical tools

Strange statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence with respect to these quantities are necessitated in my ...
1
vote
1answer
23 views

How to show a sequence of independent random variables do not almost surely converge by definition?

I have a sequence of independent random variables $X_1, X_2, \ldots$ where $$ X_n = \begin{cases} 1 & \quad \text{with probability} \ 1/n \\ 0 & \quad \text{with ...
2
votes
1answer
28 views

What is the intuitive difference between almost sure convergence and convergence in probability?

It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it ...
-1
votes
1answer
13 views

How to show convergence in probability by just using the definition?

I have a series of random variables $X_1, X_2, \ldots$ where $$ f(X_n) = \begin{cases} 1/n & \quad \text{if} \ X_n = 1 \\ 1-1/n & \quad \text{if} \ X_n = 0 \\ 0 & ...
2
votes
1answer
22 views

Weak convergence of probability measures and uniform convergence of functions

I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion: Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), ...
3
votes
1answer
11 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
-1
votes
0answers
15 views

Calculate the discrete density of the variables of a Markov chain

$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$. $Z=X+Y$ $\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition ...
1
vote
1answer
11 views

Stopping time and the Martingale stopping theorem.

According to the book that I am reading, A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of ...
0
votes
0answers
16 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
0
votes
0answers
20 views

Probability to get from A to C.

There has been a snowstorm and Bob is trying to drive from A to C. p and q are the probabilities that the two roads are passable. What is the probability that Bob can get from A to C? Note that ...
0
votes
0answers
49 views

Hypergeometric distribution with a priori probabilities of the balls

If we have a urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...
0
votes
0answers
13 views

Mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$?

Is the mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$? In particular, in the extreme case that the pairwise mutual informations are ...
0
votes
1answer
9 views

Biased voter model survival

I have a biased voter on $\mathbb{Z}^d,$ where $d>0$ (I am mostly interested in the cases where $d>1$) with the bias parameter $\lambda$. In other words, let us have a process $X=(X_t)_{t \ge ...
0
votes
0answers
14 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
0
votes
1answer
19 views

Find the joint probability density function of Max and Min

This is the problem 1.2.13 of Karlin's book An introduction to stochastic modeling: Let X and Y be independent random variables each with the uniform probability density function ...
4
votes
5answers
171 views

Finding the expected value in the given problem.

It is given that a monkey types on a 26-letter keyboard with all the keys as lowercase English alphabets. Each letter is chosen independently and uniformly at random. If the monkey types 1,000,000 ...
2
votes
0answers
6 views

(joint) Functional CLT for partial sums and counting process

Assume you are given a sequence of random variables $(X_i)_{i\geq1}$. Assume moreover that they are sufficiently smooth, say $\mathbb E[X^2]<+\infty$. Define the diffusion-scaled partial sum as ...
7
votes
3answers
230 views
+50

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
1
vote
0answers
17 views

Question on a proof involving tightness and almost sure convergence of a sequence

I'm having a hard time understanding the proof of Lemma 17 in this article. Essentially, the assertion of the lemma boils down to replacing a constant in a sequence of random variables that satisfies ...
1
vote
0answers
10 views

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
0
votes
0answers
16 views

What is the probability of unions of intersections?

Suppose we have two unions of (possibly overlapping) events. Let me denote the unions as: $$A = IE_A^1 \cup \dots \cup IE_A^{k_A}$$ $$B = IE_B^1 \cup \dots \cup IE_B^{k_B}$$ Each $IE_X^y$ is a ...
1
vote
0answers
21 views

Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
1
vote
1answer
20 views

Finding out the constant term in cumulative distribution function.

Let$$F(x)=\begin{cases} 0,\ if\ x<0\\ \frac{x^2}{10},\ if\ 0\le x<1\\\frac{x+2}{8},\ if\ 1\le x<2\\ \frac{c(6x-x^{2}-1)}{2},\ if\ 2\le x\le3\\ 1,\ if\ x>3 \end{cases}$$ Find the value of ...
2
votes
0answers
20 views

A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
0
votes
1answer
47 views

Distribution of a function of a random variable

Suppose we have continuous random variable $X$ with distribution $f_X$. That is $$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$ Now suppose I have a function $\phi: \Bbb{R} ...
1
vote
1answer
13 views

What is the distribution of $\min\limits_{1\le i\le N}\frac{X_i}{Y_i + C}$ as $N \rightarrow \infty$?

Let $Z_i = \frac{X_i}{Y_i + C}$ with $i = 1, 2, \dotsc, N$ denote the sequence of the random variables, where $X_i$ and $Y_i$ are exponentially distributed independent random variables with different ...
1
vote
0answers
24 views

Reflection principle for simple random walk

Let $(X_n)$ be a sequence of independent random variables, such that $P(X_i=1) = P(X_i=-1) = 1/2$. Then, the reflection principle states that for all $a > 0$, $$P(\max_{1\leq k\leq n} S_k \geq a) ...
-1
votes
1answer
20 views

proof for the probability$P(A^{c}\cup C)$.

Let $A,B,C$ be pair-wise independent (like A and B are independent) such that$P(A^{c}\cap B)=0.1$ and $P(B\cap C)=0.2$ Show that $P(A^{c}\cup C)\ge \frac{7}{8}$. $P(A^{c}\cup C)=1-P(C^{c}\cap ...
2
votes
0answers
20 views

Conditional expectation of another expectation expression.

What is the intuition and the proof behind the given below expression where $U,V,W$ are random variables: $E[V | W]$ = $E[E[V | U,W] | W]$ I know that $E[V | W]$ can be treated as a random variable ...
0
votes
0answers
9 views

independence of chi square distributions

We already knew that if two independent chi-squared random variables, then their sum is also chi-squared with the degree of freedoms is the sum of theirs. How about the converse? If $X\sim\chi^2(n)$ ...
0
votes
1answer
33 views

sigma algebra of a stopping time

Let $N$ be a stopping time. i.e $\{N=n\} \in \mathbb{f}_n \forall n$. $\mathbb{f}_n$ is the filtration. $\mathbb{f}_N=\{A\in \mathbb{f}, A\cap \{N=n\}\in \mathbb{f}_n \forall n\}$ is the sigma ...
1
vote
1answer
18 views

Finding bivariate probability mass function (by counting?)

Suppose that we role $d$ dice. Let $X, Y$ be random variables, where $X = \#$ rolled by the die with the highest value. $Y = \#$ rolled by the die with the second highest value. By convention, we ...
2
votes
0answers
36 views

Is it possible to upper bound conditional expectation expression: $\mathbb E[X | X > c] - c$?

As the titles suggests, I am trying to see if we can upper bound $\mathbb{E}[X \text{ }| \text{ }X > c] - c$ For now, I am assuming bounded mean on both sides: $0 < m \leq \mathbb{E}[X] \leq ...
2
votes
2answers
21 views

Prove: If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$

I'm trying to prove the following theorem using the axioms quoted below. Theorem: If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$ Axiom 1: For every event $A$ in the class ...
-1
votes
0answers
21 views

What pdf or df corresponds to this mgf? [duplicate]

Suppose we have random variables $X_1, X_2, ...$ in $(\Omega, \mathscr F, \mathbb P)$ s.t. $P(X_n = k) = \frac 1 {n+1}$ for $k = 0, 1, ..., n$ and $X := \lim \frac{X_n + 1}{n+2}$ exists. ...
2
votes
0answers
18 views

Convergence of sum dependent random variables in $L^2$ with mean zero

Suppose that $X_1, \ldots$ were a sequence of random variables with $E(X_i)=0$ and $E(X_i^2)= \sigma_i^2$ and that $S_n := X_1 + \cdots + X_n$ converged in $L^2$. Does this imply $\sum \sigma_i^2 < ...
1
vote
1answer
30 views

There is problem in calculating pgf(probability generating function)

I posted question about distribution of poisson distribution multiplied by constant. Here! From this post, i can obtain what i want. $$P(X=x)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$ $$Z=\alpha X ...
0
votes
1answer
28 views

A doubt on a proof of a theorem of Durret's Probability Theory

Below is the text of the theorem: $\mathcal{F}_{i,j}$ are sigma algebras indexed by $i$ and $j$. I'm having some difficulties in understanding this proof. Do the $\mathcal{A}_i$ contain $\Omega$ ...
2
votes
1answer
34 views

Disentangling $\int_Af(\mathbf{x})\ d\mathbf{x}$, using Fubini Theorem.

Let $\mathcal{B}^n$ be the borel sigma algebra generated by the rectangles in $\mathbb{R}^n$. I can write $f(\mathbf{x})=g_1(x_1)\cdots g_n(x_n)$. Let $\mu=\mu_1\times \cdots \times \mu_n$ be the ...
0
votes
0answers
21 views

Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...
2
votes
0answers
30 views

Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
10
votes
1answer
805 views

How do you compute numerically the Earth mover's distance (EMD)?

I was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...
1
vote
0answers
23 views

What is a Tail Field and how to interpret it?

I cannot understand or form a good intuition in my head of what a tail field is. An introduction to rigorous probability theory by Rosenthal gives the following definition: Given a sequence of events ...
0
votes
1answer
80 views

Distribution in Polya's Urn / existence of mgf / Stolz–Cesàro alternative / dominated convergence theorem

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ...
0
votes
0answers
9 views

How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf ...
2
votes
0answers
66 views

Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.

Let $\left\{ X_{n}\right\} _{n\geq1}$ be a sequence of IID random-variables s.t $\mathbb{E}\left[X_{i}\right]=0$ and $\left|X_{i}\right|\leq K$ . Let $S_{n}=\sum_{i=1}^{n}X_{i}$ , I want to show ...
1
vote
2answers
27 views

Prove that if $X_n \xrightarrow{P} c$, then $E(X_n) \to c$ for $X_n$ uniformly bounded

I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that $$E\left(X_n \right) \to c$$ ...
0
votes
0answers
15 views

Integrals of functions of statistics

Let $X: \Omega \to \mathbb{R}^n$ be a measurable random vector with law $\Lambda_X$ and probability density function (pdf) $f_X$. Let $T:\mathbb{R}^n \to \mathbb{R}^2$ a statistic (a ...
0
votes
0answers
8 views

How to find expectation of birth-death process

Let $\{X(t)\}$ be birth-death process on two-state space {0,1}. Let birth rate $\lambda=2$ and death rate $\mu=12$. How to calculate $\lim_{t\to\infty} \mathbb{E}[X(t)]$?