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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1answer
21 views

Brownian Motion Hitting Time Distribution

Define $\tau_a = \inf \left\lbrace t \geq 0 | B(t) \geq a \right\rbrace $ for some $a>0$. The problem is to show that $ \tau_a \stackrel{d}{=} \sqrt a\tau_1 $. What I've done so far: $$P(\tau_a ...
-2
votes
2answers
44 views

for a random variable $X$ with density function $f$, show that $E[g(X)]=\int_{-\infty}^{\infty} g(x) f(x) dx$ [closed]

For a random variable $X$ with density function $f$, show that $$E[g(X)]=\int_{-\infty}^{\infty} g(x) f(x) dx$$ I need some help with it by using the random variable $Y=g(x)$.
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1answer
29 views

Prove that $E[Y]=\int_0^\infty P(Y>y)dy-\int_0^\infty P(Y<-y)dy$

Prove that $E[Y]=\int_0^\infty P(Y>y)dy-\int_0^\infty P(Y<-y)dy$. Should I first prove that $$\int_0^\infty P(Y<-y)dy=-\int_{-\infty}^0 xf_Y(x) dx$$ and $$\int_0^\infty ...
3
votes
1answer
35 views

Weak convergence in probability implies uniform convergence in distribution functions

This is a problem that I am totally stuck at. I know the fact that $F_n$ converges pointwise to $F$ in this question. Also, I looked through Google and found out that I have to show first that ...
1
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1answer
37 views

thought were the same combinatorial

I was under the impression that $${52\choose 5!5!5!5!5!} = {52\choose 5}{47\choose 5}{42\choose 5}{37\choose 5}{32\choose 5} $$ Reason i ask is because i was trying to solve a simple number of ways ...
0
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1answer
25 views

probability of distinct phone numbers

the first three digits of a university telephone exchange area 452. if all the sequences of the remaining four digits are equally likely, what is the probabikity that a randomly selected university ...
0
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1answer
35 views

prove $p (\bigcup A_i) \leq \sum p (A_i) $ [duplicate]

Prove $p (\bigcup A_i) \leq \sum p (A_i) $ Now i doubt that this has not been proven already as it is a common question, but i am typing latex on my tablet because i spilled water on my computer so i ...
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2answers
28 views

transformation of single random variables with absolute value ??

integral I got the final answer to be fy(y)= 1 0< y < 1 I am not sure could anyone correct me if its wrong !
2
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0answers
30 views

Problem on exchangeable events

$\newcommand{\p}{\mathbb{P}}$ $\newcommand{\tsum}{\textstyle{\sum}}$ $\newcommand{\N}{\mathbb{N}}$ Problem: Let $\left(A_{k}\right)_{k\geq1}$ be any infinite sequence of exchangeable event, i.e., ...
2
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1answer
31 views

How to practice basic probabilistic modeling?

I'm heavily struggling in learning simple and basic probabilistic modeling. So I'm learning probability from this probability book Introduction to Probability by Dimitri P. Bertsekas. Although I ...
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0answers
22 views

problem about graph of auto-correlation for wide-sense stationary process?

I have the answers but I don't understand the idea and how it can be solved ? please clarify and help me to understand it thank you all
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
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1answer
101 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
26 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 ...
-1
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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
21 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
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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 ...
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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
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1answer
30 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
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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
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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. ...
0
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1answer
40 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
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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
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0answers
28 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
62 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
49 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
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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
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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 ...
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0answers
27 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
21 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
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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
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1answer
31 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
vote
1answer
157 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
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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
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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
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0answers
30 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
41 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 ...
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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
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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 ...
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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 ...
1
vote
1answer
15 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 ...
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 ...
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0answers
22 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 ...