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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0
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1answer
30 views

MLE of Integer Valued Normal Distribution

If Z is a normal random variable on $\mathbb{R}^d$ with parameters $(\mu,\Sigma)$ and we know that $\mu\in \mathbb{Z}^d$ and $\Sigma \in \mathbb{Z}^{d+}$; then how can we solve this MLE problem for ...
4
votes
1answer
96 views

An expectation inequality

Let $X$ and $Y$ be iid random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|]$$ Let $F(x)$ denote the distribution, after calculation, I need to prove ...
3
votes
1answer
25 views

Dose weak convergence imply tight?

$X$ is a separable metric space, $\{P_n\}_{n=1}^{\infty}$, $P$ are probability measures on $X$, and $P_n$ converges weakly to $P$, can we conclude that $\{P_n\}_{n=1}^{\infty}$ is tight? I know if ...
-3
votes
0answers
29 views

Prove that $S_n=X_1+\ldots +X_n\to \infty$ a.s. [closed]

I was stuck the problem. Can someone help me? Let $X_1,X_2,\ldots$ be i.i.d. random variables with the common density $$f\left( x \right) = \left\{ \begin{gathered} c{e^{ - ...
-1
votes
0answers
28 views

Probability of getting 3 balls in 10 rooms with 9 people [duplicate]

There is a group of 9 people who visit 10 different rooms together. Each room has 3 balls, and each person has an equal probability of getting a ball. What is the probability that, after visiting all ...
0
votes
1answer
20 views

does brownian motion and poisson random measure have to be independent? [closed]

Suppose a brownian motion $W$ and a poisson random measure $\mu$ are defined on the same filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), P)$, where both $W$ and $\mu$ are ...
1
vote
1answer
43 views

A reference for multi-dimensional characteristic functions

I'm looking for a well-written, rigorous and self-contained treatment of multidimensional characteristic functions, specifically Lévy's continuity theorem and the uniqueness theorem (which states that ...
0
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0answers
18 views

Mixture of Dirichlet Distributions

I'm working on a problem for Dirichlet distributions and I appreciate if you can give me some hints. Consider two random vectors of size K that are distributed as Dirichlet: $$\vec{Y_1} \sim ...
1
vote
1answer
28 views

Does the following sequence of random variables converge?

Let $X_1,X_2,...$ be independent random variables with $P[X_n=0]=1-1/n$, $P[X_n=1]=1/2n$, $P[X_n=-1]=1/2n$ Does $X_n$ converge almost surely? , Does $X_n$ converge in probability? I just started to ...
1
vote
1answer
55 views

If $T$ is an unbiased estimation for $X$, then is $T^2$ an unbiased estimation for $X^2$? [closed]

If $T$ is an unbiased estimation for $X$, then is $T^2$ an unbiased estimation for $X^2$? It's bugging very much. Thank you in advance!
1
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1answer
16 views

Under which conditions on $f$ do we've got $\sigma(X)=\sigma(f\circ X)$?

Let $(\Omega,\mathcal{A})$, $(\Omega',\mathcal{A}')$ and $(\Omega'',\mathcal{A}'')$ be measurable spaces $X:\Omega\to\Omega'$ be $\mathcal{A}$-$\mathcal{A}'$-measurable $f:\Omega'\to\Omega''$ be ...
1
vote
1answer
29 views

If $X:\Omega\to\Omega'$ and $f:\Omega'\to\Omega''$ are measurable and $f$ is injective, then $\sigma(X)=\sigma(f\circ X)$

Let $(\Omega,\mathcal{A})$, $(\Omega',\mathcal{A}')$ and $(\Omega'',\mathcal{A}'')$ be measurable spaces $X:\Omega\to\Omega'$ be $\mathcal{A}$-$\mathcal{A}'$-measurable $f:\Omega'\to\Omega''$ be ...
1
vote
1answer
60 views

Every zero-mean Lévy process has linear variance (wrt $t$)

I'd like to show that every Lévy process with $\mathbb{E}X_t=0, \:\forall t\ge0$ has linear variance, namely $t\mapsto\mathbb{E}X^2_t$ is linear. I showed that indeed the additivity holds, i.e. ...
-1
votes
1answer
35 views

About a probability space [closed]

Consider a probability space (Ω,A,P) and assume that the various sets mencioned below are all in A. (a) Show that if $D_i$ are disjoint and $P(C|Di)=p$ independently of i, then $P(C|⋃iDi)=p$. (b) ...
0
votes
1answer
36 views

Probability of most frequent occurrences of suits/values when drawing 4 cards from 52

Draw 4 cards from a card deck with 52 cards (4 colours and 13 values for each colour) one after the other -- none is put back. Let's have two discrete random varaibles X and Y. X counts the maximum ...
1
vote
1answer
44 views

Sum of variances of multinomial distribution.

I've k fair coins, and I would like to know the number of heads obtained in $n$ trials. But that is simple binomial distribution. But if I want to find out how much it varies from binomial ...
1
vote
0answers
26 views

Application of martingale central limit theorem

I just learned martingale central limit theorem and got a problem at hand and do not know how to form the correct martingale. Suppose we draw balls successively from a box of $2n$ balls and $n$ ...
3
votes
1answer
61 views

Prove $\limsup{A_n}\backslash\liminf{A_n}=\limsup{(A_{n+1}\backslash A_n)}$

$\newcommand{\N}{\mathbb{N}}$ Problem: Prove $\limsup{A_n}\backslash\liminf{A_n}=\limsup{(A_{n+1}\backslash A_n)}$ Attempt(Revised): (I am not sure if it's correct. I would appreciate if anyone can ...
2
votes
1answer
46 views

Calculate the probability select $k$ blue balls in box

I have a box that contains 10 balls( 2 red balls and 8 blue balls). Probability select each ball is an uniform distribution. An event is defined that selects k balls $(0<k\le 10)$ from the box and ...
3
votes
1answer
33 views

Local maximum of brownian motions

Let $B=(B_t)_{t\geq 0}$ be the standard Brownian motion. I want to show that for every $t_0 \geq 0$ $\mathbb{P}$($B$ has a local maximum in $t_0$)=0. I've already shown that for every ...
1
vote
1answer
19 views

Differentiating Spitzer's identity

Let $(S_n)$ be an arbitrary random walk. Define $$M_n:= \max(0,S_1,...,S_n)$$ and $$S_n^{+} := \max\{0,S_n\}.$$ Spitzer's identity states that for $0<r<1$, we have $$\sum_{n=0}^{\infty} r^n ...
0
votes
0answers
19 views

How to show analytically the pdf for the minimum of random variables

If $Y_1,Y_2,\ldots,Y_n$ are $i.i.d$ and each $Y_i$ is Generalized Gamma $GG(kn,\lambda)$ distributed. Assuming, the form of $Y_i = Z_i^m$ and each $Z_i^m$ is Gamma distributed, then what will be ...
1
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1answer
50 views

Image of collection of probability measures in $C_b(S)'$

Let $(S,d)$ be a Polisch space (i.e. a complete and separable metric space) and $\mathcal{P}$ the collection of probability measures on the borel sigma algebra of $(S,d)$ which we denote by ...
0
votes
0answers
26 views

Stopping and optional times.

Let $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq0},P)$ be a filtered probability space. Put $\mathcal{F}_{t^+}:=\cap_{s>t}\mathcal{F}_s$ and $\{\mathcal{F}_{t^+}\}_{t\geq0}$ be the ...
0
votes
1answer
20 views

How to compute selection probability of balls in a range

I have a question about probability that need your help. I assume that I have balls that are numbered from 1 to 100. The probability selection each ball is followed uniform distribution. I divide ...
0
votes
0answers
22 views

A proof involving conditional distributions, Poisson, binomial

Let $X$ be a non-negative integer valued random variable. Let $Y$ be the number of successes in $X$ binomial trials. Prove that, if the distribution of $Y$ and $X\mid (Y=X)$ are identical, then $X$ ...
0
votes
1answer
27 views

limsup/liminf of a random variable?

The limsup of events $A_1, A_2, ...$ is $\limsup A_n = \bigcap_{m\geq1} \bigcup_{n\geq m} A_n$ Is there a limsup for random variables $X_1, X_2, ...$? I've seen $\limsup X_n$ sometimes but it usually ...
1
vote
1answer
29 views

Is a random variable constant iff it is trivial sigma-algebra-measurable?

I found a proof here for a measurable function (instead of probability theory's random variable) being constant if and only if the sigma-algebra generated by it is the trivia sigma-algebra, I think ...
1
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1answer
155 views

Markov process on an Abelian group

Let $E$ be an Abelian group. Let $X$ be a right-continuous process with values in $(E,\mathcal{E})$ (where $\mathcal{E}$ denotes the $\sigma$-algebra on $E$), defined on $\Omega, \mathcal{F}_t,P)$. ...
2
votes
0answers
34 views

Approximating the distribution of the infinte sum of random variables

What is the formal way to show that for an infinite sum of random varaibles $\sum_{i=1}^\infty X_i$, we have $\forall\varepsilon>0,\exists N<\infty$ such that $$0<P\left(\sum_{i=1}^N X_i\leq ...
1
vote
1answer
20 views

Probability Generating Function Attempt

I am trying to find the PGF for the following distribution: $X_1$ has PMF: $\rho(x) = \frac{-p^x}{x\ln(1-p)},n\in \mathbb N$ Attempt: \begin{align*} \phi_{X}(s) &= E[s^x]\\ ...
1
vote
1answer
32 views

Gambling Game martingale

State the optional sampling theorem for martingales and bounded stopping times. You start with a capital of £100 and bet repeatedly on the toss of a coin. On each toss you may bet any whole number of ...
3
votes
0answers
28 views

Probability Generating Function of Compound Poisson Process

Let $(N_t)_{t\ge 0}$ be a poisson process with intensity $\alpha > 0$. Let $(X_n)_{n \in \mathbb N}$ be iid real valued random variables that are independent of $N_t$. Let $Y_t = ...
0
votes
1answer
38 views

Conditional expectation, max, min of random variables

We are given two independent random variables $A, B$ with uniform distribution on $[0,1]$. We define new random variables $X = \max (A,B)$ and $Y = \min (A,B)$. Find $\mathbb{E}(X\mid Y)$ (defined ...
0
votes
0answers
26 views

Absolute convergence of $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$

I want to show that if $X \in L^1$, where $X$ is a real-valued random variable, the sum $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$ converges absolute. My idea was the following: Since $X \in ...
1
vote
0answers
17 views

Right-continuous process is measurable with respect to product measure.

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{X_t\}_{t\geq0}$ be a collection of real random variables such that the map $t\mapsto X_t$ is right-continuous. Show that the map ...
1
vote
1answer
40 views

Infinite sequence of coin tosses, $\Bbb{P}(\limsup_{n\to \infty} A_n) = 1$

Let $(\Omega, \mathcal{A}, \Bbb{P}) = \otimes_{j=1}^{\infty} ( \{0,1\}, \mathcal{P}(\{0,1\}), \Bbb{P}_j)$ with $\Bbb{P}_j(1)=1/2=\Bbb{P}_j(0)$, i.e. the model for an independent infinite sequence of ...
0
votes
0answers
16 views

Martingale Dozen 1/3 statistics.

When playing on dozens there is 12/37 chance to win, and the return is 1/3. To turn odds to your favour you have to double up your money for a chance of winning money. So if i lose on my 1 unit bet, i ...
0
votes
0answers
11 views

When two random variables that have the same law… Can they be happily exchanges?

Imagine, $X$ and $Y$ are two random variables which have the same law, which we denote by $X\sim Y$. We have then a third random variable $Z$. Can we say that $$(X,Z)\sim (Y,Z)?$$ In what cases is ...
-1
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0answers
13 views

transition probability for reflected brownian motion [closed]

I have to show that the transition probability for the reflected Brownian is given by $p_+(t,x,y)=[p(t,-x,y)+p(t,x,y)]$ where $p_+=\frac{1}{\sqrt{2\pi t}}e^{-|x|^2/2t}$ and also show that it is a ...
2
votes
1answer
19 views

Scenario in which Dominated Convergence does not hold

For $X,X_n,Y \in \mathscr L^1, n\in \mathbb N$ such that $\lvert X_n \rvert \le Y, Y \ge 0$. By the dominated convergence theorem: $$ \lim_{n \to \infty} E[X_n] = E[X] $$ and can be proven by Fatou's ...
0
votes
0answers
21 views

To prove that induced probability measure indeed defines a probability measure

Given two measurable spaces $(Ω_1, B_1)$ and $(Ω_2, B_2)$, a measurable function T : $Ω_1 → Ω_2$ and P is a probability measure on $(Ω_1, B_1)$. $B_1$ & $B_2$ are respective sigma algebras. The ...
0
votes
1answer
39 views

Rolling a Fair dice 6 times

If I roll a fair dice 6 times, which event is more likely A) 1,1,1,1,1,1 B) 3,2,2,5,1,6 The probability of (A) occuring is [1/6]^6 = 2.14E-05. I'm not too sure how to calculate (B)'s probability ...
0
votes
0answers
21 views

Are convex functions of a random variable themselves random variables?

I was looking at proofs of Jensen's inequality and noticed that they usually assume that for a convex function $g$ and a random variable $X$, the expression $\mathbb{E}(g(X))$ is well defined, which ...
0
votes
0answers
22 views

Law of iterated logarithm for Markov Chains

Does anyone know where(or if) I can find a proof of law of iterated logarithm for irreducible and aperiodic Markov chain with finite number of states. All of the proofs I have seen so far are really ...
1
vote
1answer
40 views

Show limsup/liminf is in tail field

Given events $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_n, X_{n+1}, ...)$ be their tail field. 1 How do I show that $\limsup X_n \in \tau$? What I tried: $\limsup X_n = ...
3
votes
1answer
18 views

Proof of uniqueness of conditional expectation

I have a question on the proof Durrett (p. $190$) gives for the uniqueness of the conditional expectation function. If I understand his proof correctly, here is what I think it is saying: Suppose ...
1
vote
2answers
44 views

Constructing measure preserving maps between non-atomic measures

Suppose $(\mu, X,\Sigma)$ and $(\mu^\prime, X^\prime, \Sigma^\prime)$ are non-atomic probability measures. Is it always possible to construct a measure preserving map between the two spaces? (If ...
1
vote
1answer
17 views

Finding linear orders on a measure space whose initial segments have all possible measures

Let $\mu$ be a non-atomic probability measure on some space $(X, \Sigma)$. Is it always possible to find a linear order, $\leq$, on $X$ such that $\mu: \mathcal{A}\rightarrow [0,1]$ is surjective, ...
0
votes
2answers
33 views

$E[X(X-1)]$ of a Binomial random variable

Let $X~\sim B(n,p)$ be a binomial random variable. Calculate $E[X(X-1)]$. Do I need to use the Binomial theorem? If yes, how?