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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28 views

Probability of Intersection between $B$ and NOT $A$

There are two events, $P(A) = .45$ & $P(B) = .65$. Also, $P( A \cap B) = .25$. How can I get $P (B \cap A^c)$ ? Any help will be appreciated. Thanks!
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2answers
118 views

N Identical Balls numbered through 1 to n

A box contains n identical balls numbered 1 through n. Suppose k balls are drawn in succession. What is probability that m is largest number drawn ? What is the probability that the largest number ...
3
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2answers
54 views

the set of points where f is continuous is in Borel

For a function $f:[0,1]\to \mathbb{R}$, let $C$ be the set of points where $f$ is continuous. Prove that $C$ is in the Borel $\sigma$-algebra. I know that for $A=\{f(x): f(x)<a\}$ is open for ...
0
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1answer
43 views

Showing That a Certain Sequence of Random Variables is i.i.d.

I am working on the following problem for my probability theory course. Let $U$ be a Uniform($[0,1]$) random variable (i.e., the distribution of $U$ is the Lebesgue measure on $[0,1]$). Define ...
0
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1answer
31 views

Proof that Lévy metric is symmetric

Let $F$ and $G$ be distribution functions and define the Lévy metric by, $$ d_L(F, G) = \inf \{\epsilon > 0\ |\ G(x - \epsilon) - \epsilon \leq F(x) \leq G(x + \epsilon) + \epsilon {\rm\ for\ all\ ...
1
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2answers
32 views

Smallest no of balls in the box?

A box contains white and black balls. When two balls are drawn without replacement suppose the probability that both are white is 1 /3. a) Find smallest number of balls in the box ? b) How small ...
5
votes
1answer
171 views

$\limsup_{t \to 0} {L_t}/\sqrt{t} = \infty$ with probability one?

Let $B_t$ be a standard Brownian motion, $L(x, t)$ be the local time $x$ at time $t$, and $L_t = L(0, t)$. Do we have$$\limsup_{t \to 0} {{L_t} \over{\sqrt{t}}} = \infty$$with probability one?
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2answers
44 views

Inequality for Limit inferior

I was wondering if anyone can give a mathematical explanation for the claim below. If $F(x-\epsilon)\le F_n(x)+p_n$ where $\lim_{n\to\infty}p_n=0.$ (1) Then $F(x-\epsilon)\le ...
2
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1answer
53 views

$\mathbb{P}\{B_2 > 0 \text{ }|\text{ } B_1 > 0\}$ [closed]

What is$$\mathbb{P}\{B_2 > 0 \text{ }|\text{ } B_1 > 0\},$$where $B_t$ is a standard one-dimensional Brownian motion?
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2answers
34 views

Problematic exercise on alternate way of expressing random variables

I found this exercise on a book on probability theory, and I find it problematic. Let $(X, \Sigma, p)$ be a probability space, $\mathcal{A} \subseteq \Sigma$ a finite partition of $X$; and $\phi ...
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1answer
34 views

Constant $c$ such that $\lim_{x \to \infty} xe^{x^2/2} \textbf{P}\{N \ge x\} = c$?

Let $N$ denote a $N(0, 1)$ random variable. What is the constant $c$ such that$$\lim_{x \to \infty} xe^{x^2/2} \textbf{P}\{N \ge x\} = c?$$
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2answers
28 views

Why my expression for coupon collector is giving wrong output

y=n*log(n)+0.50*n+0.50; This is my expression but it is giving 36.32 for n=12 while it has to give 37.24/ edit:This is the problem ...
0
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1answer
24 views

MC whose transition probabilities have a PDF w.r.t. $\mu$ are reversible w.r.t to $\mu$

Let $(Y_n)_{n \in \mathbb{N}}$ be a Markov Chain with transition probability $$p(x, dy) \sim N(x, \epsilon)$$ Show that $Y$ is reversible w.r.t to the lebesgue measure . What I have done is just ...
5
votes
1answer
42 views

Show $(\mathbb{E}\vert X^2 - \mathbb{E}[X^2]\vert)^2 \leq 4\mathbb{E}[X^2](\mathbb{E}[X^2]-\mathbb{E}[X]^2)$

Anyone have any leads on this? X has a finite second moment and is nonnegative. \begin{equation} (\mathbb{E}\vert X^2 - \mathbb{E}[X^2]\vert)^2 \leq 4\mathbb{E}[X^2](\mathbb{E}[X^2]-\mathbb{E}[X]^2) ...
0
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1answer
34 views

random vector density

Consider now a vector $X = (X_1, X_2)$ where $X_1$ and $X_2$ are real-valued random variables. Let $X_1$ and $X_2$ have densities $f_1$ and $f_2$ on $\mathbb{R}$. Show by example that it is possible ...
1
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1answer
32 views

$L^2$ convergence from convergence in distribution and uniform integrability

Is it true that if $X_n \to X$ in distribution and $\{X_n^2\}$ are uniformly integrable then $X_n \to X$ in $L^2$
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2answers
25 views

Normal distributions with bounded means and variance are tight

Let $(a_n)_n$ and $(\sigma_n^2)_n$ be sequences of real numbers with $\sigma_n^2>0$ for all $n\in \Bbb N$ and let $$\mathcal P = \{\Bbb P^{X_n} : X_n \sim \mathcal N (a_n,\sigma_n^2)\}.$$ Then ...
0
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0answers
43 views

Does a critical point of log-normal distribution exist?

To find a critical point(s) of a multi variable function we need to find the first partial derivative with respect to each variable, set each to zero and solve the system. We can then use the second ...
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2answers
31 views

How to understand independent events' set structure

$\Omega_1=\{A,B,C\} \\ \sigma(\Omega_1)=\{\varnothing,\{A\},\{B\},\{C\},\{A,B\},\{A,C\},\{B,C\},\{A,B,C\}\} \\ \mathsf P_1(\{A\})=1/3,\mathsf P_1(\{B\})=1/2,\mathsf P_1(\{C\})=1/6 \\ \Omega_2=\{D,E\} ...
0
votes
1answer
117 views

Probability of a point in the unit square being in the interval (a,b)

Let $(X, Y)$ denote a uniformly chosen random point inside the unit square, $$[0,1]^2=[0,1]\times[0,1]=\{(x,y):0\le x\le1,\ 0\le y \le1\}$$ (a). Let $ 0\le a\lt b\le1$ find the probability $P(a\lt ...
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0answers
53 views

Y is $\sigma (X)$-measurable, show $Y=f(X)$ $f$ a Borel function

I am working on the following question for my probability theory class: Let $X, Y$ be random variables on $(\Omega,\mathcal{F}, P)$ such that $Y$ is $\sigma (X)$-measurable. Show that there is a ...
1
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2answers
33 views

About a identity in probability theory.

Let $(\Omega,\mathcal{F},P)$ be a probability space, and $\{A_n\}_{n=1}^{\infty}$ a sequence of subsets in $\Omega$ such that: i) $A_n \in \mathcal{F} \hspace{0.5cm} \forall n=1,2,...$ ii)$A_1 ...
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0answers
20 views

Expectation and Stochastic Dominance

Given two distributions $F_1(x)$ and $F_2(x)$. I know that the following two expectations are equal. $$ \int_x \frac{a}{c-x}dF_1(x) = \int_x \frac{b}{c-x}dF_2(x) = C>0 $$ And I know that ...
3
votes
2answers
143 views

Finding an error estimation for the De Moivre–Laplace theorem

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De ...
0
votes
1answer
31 views

Let $X \sim Bin(n,p)$ show that $X/n \nsim Bin()$

Let $X \sim Bin(n,p)$ Part A. Show that the argument "Then $X/n \sim Bin()$ with $E[X/n]=p$and $Var[X/n]=pq/n$" is false. My book says we can prove this result using a moment generating ...
0
votes
1answer
28 views

Is my proof for showing $P(X=k|Y=n)$ follows $B(n,1/2$) sufficient? with $Y$ a Poisson law, $X$ modelling a number of iid events

I've already worked on a couple of random variables with their value in $\mathbb{N}$ $(X,Y)$ $p_{kn}=P(X=k,Y=n)= \left( \frac{λ^ke^{-y} α^n(1-α)^{k-n}}{n!(k-n)!} \right)\mathbb{1}_{\{0\le n\le k}\} ...
1
vote
1answer
63 views

Prove if $E[Y|\mathcal{G}] = X$ and $E[Y^2|\mathcal{G}] = X^2$ then $X=Y$ almost surely

This is the complete problem: Let be $(\Omega, \mathcal{F}, P)$ probability space. $\mathcal{G}$ is a sub $\sigma$-algebra of $\mathcal{F}$. X and Y are random variables with Y square integrable. ...
0
votes
1answer
32 views

Find posterior coin tossing

Suppose that $Y$ is the number of heads in $n$ tosses of a coin, binomially distributed with index n and parameter $\theta$ and that the prior distribution of $\theta$ is of Beta form with density ...
1
vote
1answer
26 views

Law convergence and Levy

Let $X_n$ be a random variable that converges weakly to some $X$. In my case $X\sim N(0,1)$ I want to prove that $E[e^{kX_n}] \to E[e^{kX}]$ for some $k \in \mathbb{N}$. I've heard that this is a ...
1
vote
1answer
48 views

Application of optional sampling theorem in a proof of the Burkholder-Davis-Gundy inequalitiy

(Doob's optional sampling theorem ):Let $(X_t)_{t \in \mathbb{R}_+}$ be a right continuous martingale and $\tau \text{ and} \sigma $ be two stopping times such that $\sigma \leq \tau$ a.s. If we ...
1
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0answers
56 views

The stick-breaking scheme

Consider a stick-breaking scheme with independent variables $Y_k$ with $$1−P\left(Y_k = 0\right) = \frac{1}{k^2} = P\left(Y_k = 1 − e^{-k}\right).$$ I want to show that stick is “not finished”: ...
0
votes
1answer
50 views

Convergence in probability of exponentially distributed random variable

I have the following problem: Let $(X_{k})_{k\geq1}$ be a sequence of independent, exponentially distributed random variables with $\lambda=1$. Show that for all $\alpha<1$, we have ...
2
votes
1answer
49 views

Mean, variance and nothing else

Is there a distribution which has non-zero variance, but all higher central moments being zero? If not, what is the easiest (less technical) way to show that? Does similar result holds if we talk ...
0
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0answers
18 views

A question involving kernels on measurable spaces

Let $(E, \mathcal{B}(E)), (F, \mathcal{B}(F))$ measurable spaces. A $\it{kernel}$ from $(E, \mathcal{B}(E))$ to $(F, \mathcal{B}(F))$ is a map $N : p\mathcal{B}(E) \to p\mathcal{B}(F)$ such that: $$N ...
7
votes
1answer
151 views

Density given by variable-coefficient PDE

I am looking for a time-dependent probability density $f(x,y,t)$ solving the equation $$-\frac{\partial f}{\partial t} = \alpha\cdot \big(y - F(x)\big)\frac{\partial f}{\partial x}+\beta\cdot ...
0
votes
1answer
48 views

Joint probability distribution from all conditionals. Why is it not possible?

My intuition just led me completely wrong. Given the fact that Gibbs sampling is working, why isn't it always possible to calculate the joint probability distribution? (Feasibility aside) Gibbs ...
2
votes
0answers
27 views

What is the relevance of the definition of a closed submartingale

In my course on Stochastic Calculus,a closed submartingale was defined Def:We say that a submartingale $(X_t)_{t \in \mathbb{R_+}}$ is closed by $X_{\infty}$ if there exists a random variable ...
2
votes
1answer
44 views

Show that $f(x) = \mathbb{E} e^{(-\vert Y - x\vert)}$ is continuous on the real line, where $Y$ is a random variable

I'm having a hard time getting started with this one. Any hints? Show that $f(x) = \mathbb{E} e^{(-\vert Y - x\vert)}$ is continuous on the real line, where $Y$ is a random variable. Attempt: Let ...
2
votes
1answer
53 views

A.s. convergence of densities implies convergence in distribution?

Problem. Let $X,X_1,X_2,...$ be random variables with distribution functions $F, F_1, F_2,...$ and $\lambda_1$ densities $f, f_1,f_2,..$ respectively. Is it true that $$\lim_{n\to \infty} f_n = f ...
0
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0answers
41 views

Compute $Pr[Geo(1/8) < Geo(1/4) ]$

Let $A \sim Geo(1/8)$ and $A \sim Geo(1/4)$ be two independent variables. Compute $Pr(A<B)$. Attempt: Let $Z = A-B$. We seek $$P(Z>0)= \sum_{z=1}^{\infty} P(Z=z).$$ And $$P(Z=z) = \sum_{k = ...
2
votes
1answer
43 views

For i.i.d. $X,Y\sim \mathcal N(0,1)$ we have $\max(X,Y)\sim \frac{(|X|+Y)}{\sqrt 2}.$

Problem. Let $X,Y$ be independent and identically $\mathcal N(0,1)$ distributed random variables. Then $$\max(X,Y)\sim \frac{(|X|+Y)}{\sqrt 2}.$$ My attempt: This is supposed to be an exercise ...
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0answers
25 views

The inner product spaces and linearity in probability

Consider a class of inner product spaces $$\langle \cdot,\cdot\rangle_{{\lambda}\in \Lambda}: R^n\times R^n\to R$$ parameterized by $\lambda \in \Lambda=\Delta(\{w_1,....,w_n\})$, the set of all ...
3
votes
1answer
57 views

computation of the probability of a random variable

$(Y_n)_n$ is a sequence of random variable i.i.d such that $Y_1=1$ with probability $p$ and $Y_1=-k$ with probability $1-p$. $(S_n)_n$ is a sequence of random variables defined as $S_0=0$ and ...
1
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0answers
26 views

Prove $\lim_{n\to\infty}\Bbb P[\max_{k\le n}x_k \le\sqrt{2\log n-\log(2\log n)-log 2\pi +2x}]=\exp({-e^{-x}})$

Prove that $\lim_{n\to\infty}\Bbb P[\max_{k\le n}x_k \le\sqrt{2\log n-\log(2\log n)-log 2\pi +2x}]=\exp({-e^{-x}})$, where $x_1, x_2$, etc. are independent with common density ...
1
vote
0answers
30 views

Upper bound for the $t$-th moment in terms of lower moments

Let $a_1, \ldots, a_n$ be positive integers. For positive integers $t$ and $m$ define the sum $$ M_t(m) = \dfrac{1}{n} \sum_{k=1}^n |a_k - m|^t. $$ I'm interested in upper bounds for $M_t(m)$ in ...
2
votes
1answer
70 views

Proving convergence in quadratic mean of a sequence of random variables [closed]

Let $X_1, X_2,\ldots$ be a sequence of random variables. Show that $X_n$ converges to $b$ in quadratic mean if and only if $$\lim_{n\to\infty}\mathbb E[X_n] = b $$ and ...
2
votes
1answer
48 views

From Cauchy in Measure to Almost Sure Convergence

My setup is the following. Consider a sequence of random variables $(X_n)_n$ such that for every $\delta>0$ \begin{align*} \lim_{n,m\rightarrow\infty}\mathbb{P}[\sup_{m<k\leq n}|X_m-X_k|\geq ...
0
votes
2answers
27 views

Expectation of a Product of Independent Random Variables

I was wondering if there was a formula for the expectation of a product of $n$ independent random variables. I have only seen one for two random variables. I guess what I am asking is: Let $X_1, ...
0
votes
0answers
30 views

Conditioning formula for event with Probability Zero

Recall that for events $A,B$ and $C$ with $P(C)>0$, $$ P(A \cap B \mid C) = P(A \mid B \cap C)\cdot P(B \mid C). $$ I'd like to show an analogous result when $P(C) = 0$ but am having trouble. The ...
0
votes
0answers
70 views

What does it mean to “describe” an event in probability?

This may be too specific of a question, and if so I apologize, but what does my probability textbook mean when it asks that I "describe" events? Here is the question: Two dice are thrown, let E be the ...