1
vote
0answers
12 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ b) Prove that $ X_n ...
0
votes
1answer
19 views

Equivalent definition of random variables

I've come across the following two definitions of random variables and am trying to figure out if they are equivalent or not. Let $\Omega$ denote our sample space and $\mathscr{F}$ denote our ...
0
votes
3answers
79 views

Prove that $\|x+y\|_{\infty} \leq \|x\|_{\infty} + \|y\|_{\infty}$.

Suppose $\left(X, \Sigma, \mu \right)$ is a measure space and $x,y \colon X \longrightarrow \mathbb{R}$ are random variables. We define $$\|x\|_{\infty} := \inf_{A \subseteq \Sigma, \mu(A)=0} ...
0
votes
1answer
33 views

The characteristic function of the random time $N$

The rv's $X_1,X_2,X_3,\ldots,X_n$ are I.I.D and have the following pmf's: $$p_x(-1)=1/4\quad p_x(0) = 1/2,\quad p_x(1) = 1/4$$ The random time $N$ is defined as: $$N = \min\{n \mid X_n = 0\}$$ ...
2
votes
1answer
22 views

Infinitely many “records” of uniform random variables

I am doing the following exercise: Let $(U_n)_{n \geq1}$ be iid uniform random variables on $[0,1]$. Define the event $E_n = U_n>\max \lbrace U_1, U_2, \dots, U_{n-1} > \rbrace$. I.e. the ...
3
votes
1answer
29 views

General Weak Law of Large numbers

I came across a question regarding the WLLN. Suppose for $X \geq 0$ , $\mathbb{E}[X] = \infty $ , $S_n = \sum_{i \leq n} X_i$, $X_i$ are iid copies of $X$ , and $\frac{\mathbb{E}[X \mathbf{1} _{X ...
1
vote
1answer
21 views

A problem on almost sure convergence of an average

I have the following exercise: Let $X_1, X_2 \ldots$ be such that $$ X_n = \left\{ \begin{array}{ll} n^2-1 & \mbox{with probability } n^{-2} \\ -1 & \mbox{with probability } ...
-1
votes
1answer
43 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
0
votes
1answer
16 views

Using Monotone Convergence Theorem to extend a result involving random variable

We assume that for a non-negative, bounded, continuous random variable we have $$ E[X]=\int_0^\infty P(X>x) dx $$ Now the task is to extend this result to non-negative, continuous random variables ...
0
votes
0answers
26 views

Convergence of random variables depending on the measure

Suppose the probability spaces $\left([0,1], \mathcal{B}([0,1],\mu_i \right)$ for $i=1,2,3$ , where $$ \mu_1 = \lambda , \ \ \ \ \ \ \ \ \ \ \mu_2 = \delta_1,\ \ \ \ \ \ \ \ \ \ \mu_3 = ...
-1
votes
1answer
33 views

sums and distance of uniform distributions

Let $X$ and $Y$ be two uniformly distributed, independent random variables on the interval $[0,b]$. Let $S = X+Y$ be their sum and $D = |X-Y|$ be their distance. I have a few questions: a) To ...
0
votes
3answers
60 views

Why is a probability density function nonnegative?

Let $X$ be a random variable and its density $f$ be defined to be the derivative of its distribution function $F$, i.e. $$\Pr(a< X\le b)=F(b)-F(a)=\int_a^bf(x)\operatorname{dx}$$ Now let ...
2
votes
1answer
60 views

Showing that $(1-u)z^2\leq P(uz\leq |X|)$ when $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$.

I am trying to show that $$(1-u)z^2\leq P(uz\leq |X|)$$ where $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$. I've been given a hint to consider Cauchy-Schwarz, however, I don't see where ...
0
votes
1answer
16 views

How can we derive cross covariance $R_\mathrm{xy}(t_1,t_2)=R_\mathrm{yx}^*(t_2,t_1)$?

In random process, cross covariance is nonnegative definite like $$R_\mathrm{xy}(t_1,t_2)=\mathbf{E}(\mathrm{X}(t_1)\mathrm{Y}^*(t_2))=R_\mathrm{yx}^*(t_2,t_1)$$ I'm wondering how it can be derived. ...
0
votes
1answer
33 views

The quantile function $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$ has the condition that $Pr(X=c) = p_1 - p_0$

Let $X$ be a random variable with c.d.f. $F$ and quantile function $F^{-1}$. Assume the following three conditions: (i) $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$, (ii) ...
0
votes
1answer
65 views

PDF and CDF of probability theory [closed]

The continuous random variable X has pdf $$f(x) =\begin{cases} x/2, \ 0<=x<=2 \\ 0, \ \text{elsewhere} \end{cases} $$ Two independent determinations of X are made. What is the probability ...
2
votes
1answer
37 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all ...
1
vote
1answer
37 views

limsup of a sequence of random variables (definition)

Let $X_n$ be a sequence of random variables. First, $\limsup X_n=\inf_n\{\sup_{m\ge n}X_m\}$. So, $$\{\limsup X_n\le c\}=\bigcup_n\bigcap_{m\ge n}\{X_m\le c\}$$ Is it correct to say that if ...
0
votes
1answer
34 views

Proving Isserlis' Theorem for n=4

I have been trying very hard to prove Isserlis' theorem for n=4 case, i.e when we have 4 random variables that are jointly Gaussian variables with zero-means. ...
0
votes
0answers
14 views

Show Y is location-scale if $\sigma > 0$ is unknown

Let X be a random variable having the gamma distribution with shape parameter $\alpha$ and scale parameter $\gamma$, where $\alpha$ is known and $\gamma$ is unknown. Let $Y= \sigma $ log $X$. Show ...
1
vote
2answers
44 views

Does $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ imply $\phi(X_n) \xrightarrow{\mathbb P} \phi(c)$ in this case?

Let $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ and $\phi: \mathbb R \to \mathbb R$ be bounded, continuous in $c$, and $\phi(c)=0$. Show that $\mathbb E\left[\phi(X_n)\right]\to0.$ I was going ...
0
votes
1answer
22 views

Joint Density function from marginal density functions

This there anyway to find the joint density function of random variables X and Y. Nothing is given about they being independent. So we have to solve by assuming that they are not independent.The ...
1
vote
2answers
43 views

Does $X_n \xrightarrow{\text{in distr.}} X$ and $|X_n|\leq Y$ imply $|X|\leq Y$?

We know that $$X_n \xrightarrow{\mathbb P} X \text{ and } |X_n|\leq Y \implies |X|\leq Y \text{ a.s.}$$ I was wondering if the same holds in case of convergence in distribution. So far, I've shown ...
0
votes
1answer
26 views

When is a random variable is said to be well-defined?

In the paper On the Bootstrap of the Sample Mean in the Infinite Variance Case by Keith Knight, on page 1170 at the bottom of the page before the theorem, the author mentions that the random variable ...
1
vote
1answer
31 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
0
votes
1answer
60 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
0
votes
1answer
29 views

Corollary of Kolmogorov zero-one law

Here is another corollary of the theorem: Kolmogorov zero-one law given in my textbook (Probability path). How can I apply the said theorem given that if $X_n$ are independent random variables, ...
3
votes
1answer
45 views

Construct independent $X_n$ such that $\sum_{n=1}^\infty Var(X_n)=\infty$

How to construct an independent random variable $(X_n)_{n\in\mathbb{N}}$ such that $\sum_{n=1}^\infty X_n$ converges and $\mathrm{Var}(X_n)$ is uniformly bounded by some constant C, but ...
3
votes
0answers
38 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
2
votes
2answers
34 views

Definition of multivariate random variable

Let $(\Omega,\mathcal E,P)$ be a probability space and $X_1,\dots,X_n$ random variables on $(\Omega,\mathcal E, P)$. Then I can define the vector $X=(X_1,\dots,X_n)\colon \Omega \to \mathbb R ^n$ and ...
0
votes
0answers
23 views

Convergence of random saddle point

Let $y_n^*$ be the solution of $$ y = g_n(y) $$ where $g_n(\cdot)$ is a random function. Suppose that for fixed $y$ $$ g_n(y)\to h(y) $$ almost surely and pointwise as $n\to\infty$. Is there any ...
0
votes
1answer
58 views

Chebyshev Inequality

I am reading a research paper, and the author claims to get to a desired result by making use of the Chebyshev Inequality. I can get to the desired result also with some reasoning, but I fail to ...
1
vote
0answers
20 views

Bounding the Correlation Coefficient

Let us assume we have two random variables $X$ and $Y$ where $X = f(A, B, C)$ and $Y = g(A, B, C)$. $A, B, C$ are 3 independent random variables and the functions $f, g$ are known but rather expensive ...
0
votes
2answers
40 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
1
vote
1answer
13 views

Notation for image of a discrete random variable?

Suppose we have a discrete probability space $(\Omega,\Sigma,\mathbb{R})$ and a discrete random variable $X:\Omega \to \mathbb{R}$. A usual way to denote the set of values that $X$ takes is simply ...
0
votes
1answer
40 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
1
vote
0answers
34 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
1
vote
2answers
43 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
0
votes
0answers
49 views

If $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ can we explicitly define the r.v. $(Y|X\in A)$?

When introducing conditional expectation, one can define $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ which is itself a law. I was wondering if there is a way to define a random variable ...
2
votes
1answer
63 views

If $X$ is Poisson, find the expectation of $\frac{1}{a+X}$

If $X$ is a Poisson random variable with $\Pr(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$ and $a>0$ then find the expectation of $\frac{1}{a+X}$ If I make use of ...
-1
votes
1answer
35 views

finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
3
votes
1answer
15 views

$L^p$ integrability of products of Gaussian variables

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm: $$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot ...
-1
votes
2answers
48 views

How to work with the mode of a probability mass function

How do you work with a probability mass function in determining stuff related to the mode. Here's the question I have $P(X=x) = {\theta^n}{{n}\choose{x}}({\frac{1-\theta}{\theta}})^x, x = ...
1
vote
1answer
24 views

A LLN type theorem on the supremum of functions of a RV

Let $X_1,\dots,X_n$ be iid real valued random variables. Let $\mathcal{F}$ be a set of functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\mathbb{E}f(X_i) < \infty$ for all $f \in ...
2
votes
1answer
61 views

If $Y\ge 0$ almost surely and $X+Y \sim X$ then $Y=0$ almost surely

Let $X, Y$ be random variables on the same probability space such that $Y \ge 0$ almost surely and $X+Y$ and $X$ have the same distribution. Please resolve whether these conditions imply that ...
1
vote
1answer
40 views

Conditional expectation of symmetric Sigma algebra

Another exercise with conditional expectation that I have problems with. Let $\Omega=[-1,1]$, $\mathcal{F}=\mathcal{B}(\Omega)$, $\mathbb{P}=\frac{1}{2}\lambda$. Let X be a ...
5
votes
2answers
42 views

Finding all Borel measures $\mu_X$ such that $Y\sim \mathcal{N}(0,1) \Rightarrow XY \sim \mathcal{N}(0,1)$.

Find all Borel measures $\mu$ on $\mathbb{R}$ such that for every independent random variables such that $X \sim \mu$ and $Y\sim \mathcal{N}(0,1)$ we have $XY \sim \mathcal{N}(0,1)$. To be honest ...
1
vote
0answers
35 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
2
votes
0answers
40 views

Almost sure limit of $\log(X_1 + X_2 + … + X_n) - \log(n)$

Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$ Would it ...
1
vote
2answers
16 views

Geometric and binomial distribution problem

Let $X \sim Bi(n,p)$, and $Y \sim \mathcal{G}(p)$. (a) Show that $P(X=0)=P(Y>n)$. (b) Find the number of kids a marriage should have so as the probability of having at least one boy is $\geq ...