Questions about maps from a probability space to a measure space which are measurable.

learn more… | top users | synonyms

1
vote
1answer
42 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
0
votes
1answer
63 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
4
votes
2answers
48 views

Conditional Expectation of X given X^2

What can we say about $E[X|X^2]$ in general? And if $X$ has density $f$ respect the Lebesgue measure?
0
votes
1answer
25 views

$\sigma(Y)$-measurable R.V. $X$ and Borel functions

I have to prove that if $Y: \Omega \rightarrow \mathbb{R}$ then $X: \Omega \rightarrow \mathbb{R}$ is a $\sigma(Y)$-measurable function if and only if exists a Borel function $f: \mathbb{R} ...
1
vote
0answers
38 views

Moment-generating function of a generalised normal random variable

Let $X$ be a random variable that follows the "version 1" generalised normal distribution described here, with p.d.f. ...
0
votes
2answers
27 views

Probability of special configuration of ones in a binary string

Consider the sequence $(X_i)_{1 \leq i \leq L}$ of i.i.d. random variables, where $X_1 \in \{0,1\}$ and $P(X_1 =1) = p$. For a $k \in \mathbb{N}$ define the event $A_{k,L}$ as "all ones in the ...
0
votes
0answers
13 views

Two i.i.d Rvs (Gaussian)

Q: You have two i.i.d Rv's X~N(0,1) Y~(0,1). Let Z=(X+Y)^2. a) Find the mean on Z i.e E[Z}. b) Find Corr(X,Z) & Corr(Y,Z). c) Determine if Z & Y are uncorrelated. Ans: Finding E[Z] was ...
1
vote
3answers
55 views

Random Variable Problems?

Can someone show me how to work this out? I can't get the answers in the boxes.
2
votes
2answers
65 views

Almost sure convergence of a sequence of random variables

Once again I've encountered a problem, which might not be difficult: I'm given a sequence of random variables $ (X_n) $, each with density function $g_n(x) = nx^{n-1} \textbf{1}_{(0,1]} $. I am to ...
3
votes
1answer
169 views

PDF of sum of two random variables

Assume an $n$ dimensional random variable $U$ that is uniformly distributed in the volume of an $n$-sphere with radius $R$. Assume another $n$ dimensional random variable $N$ that is distributed ...
1
vote
1answer
43 views

Convergence in probability of iid normal random variables

Let $X_1, X_2,\ldots$ be a sequence of iid normal random variables with zero mean and unit variance. I read the following as a trivial example: (1) $X_n \to X_1$ in law, (2) $X_n \not\to X_1$ in ...
2
votes
0answers
27 views

Rosanov - Probability Theory Chapter 4 Question 5

I am trying to solve one of the questions in Rosanov - Probability (Chapter 4 Question 5), but I am not exactly sure what the question is asking of me. The question is: Random variable $E$ with ...
0
votes
0answers
17 views

The distribution of minmax and maxmin deviations of a Random variable

Let $X_1,X_2,X_3,......,X_n$ be $n$ independently and uniformly distributed random variables in the interval $[a,b]$. Further let $P=\min \{X_i,i=1,2,3..,n\}$ and $Q=\max\{X_i,i=1,2,3..,n\}$. ...
-1
votes
1answer
19 views

Finding number of points in a bounded set when number of points in the unbounded set are known.

Consider a random distribution of points in a Random 2D plane. I would like to find the number of points in a circle within this plane. Can anybody helps in solving the problem? Regards
1
vote
1answer
26 views

Borell Cantelli Application

If i got that $\mathbb{P}(\underbrace{|X_{n}|>n^{\frac{1}{2}+\epsilon}}_{=:A_{n}})\leq \exp\left(-\frac{n^{2\epsilon}}{8}\right)$ with $\epsilon \in (0, 0.5)$. I know that ...
1
vote
1answer
38 views

Random variables $x_i$ with $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$

I am looking for a sequence $(x_n)_{n\in\mathbb N}$ of random variables such that the sequence hasn't any expected value and $\lim_{k\to\infty}\frac1k\sum_{i=1}^kx_i=0$. I thought about using a ...
3
votes
2answers
179 views

Variance of a function of independent random variables

Suppose I have two discrete independant random variables $X$ and $Y$, and that I'm interested in the expected value of the random variable $W$, where: $$ W= \text{sign}(X-Y). $$ So, W is 1 if ...
0
votes
0answers
23 views

Weighted random walk in 1-dimension

Suppose we have random walker on a line, he can only stay on sites which are, say, a distance $a$ from each other. At each step he can go left or right. Every time he steps on a site, makes the ...
2
votes
0answers
34 views

$\sup_nX_n<\infty$ almost surely iff $\sum_nP(X_n>A)<\infty$

Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables. Show that $\sup_nX_n<\infty$ almost surely iff there exists $A>0$ such that, $\sum_nP(X_n>A)<\infty$ By ...
1
vote
0answers
34 views

How do I prove the special case of the central limit theorem?

Let $(X_n)$ be an i.i.d. sequence such that $\mathbb P(X_1=1)=\frac{1}{2}+\varepsilon$ and $\mathbb P(X_1=-1)=\frac{1}{2}-\varepsilon$ for some $\varepsilon\in(0, \frac{1}{2})$. I'd like to show that ...
1
vote
1answer
36 views

Prove $\{Z=0\}\subset\limsup\limits_{n}\{X_n<\epsilon\}$

Let $(X_n)_{n\in\mathbb N}$ be independent real random variables, with values in $(0,\infty)$. Consider the random variable $Z(w):=\inf\limits_{n\in\mathbb N}X_n(w)$. Prove that for every fixed ...
2
votes
1answer
38 views

Finding the distribution of $M$

Let $(X_1,X_2)$ be uniformly distributed in $[0,1]^2$ and define $Y_1=\max(X_1,X_2)$, $Y_2=\min(X_1,X_2)$. What is then the distribution of $M:=Y_1-Y_2$ ? To find the joint distribution we take a ...
1
vote
2answers
95 views

Density function for a random variable having a mixed distribution

A random variable has the following mixed distribution (ie: A distribution that is both discrete and continuous): $P_{X}=\frac{1}{3}E(1)+\frac{2}{3}B(\frac{1}{2})$ Where E(1) is the exponential ...
1
vote
1answer
23 views

Convergence of random variable

I've been facing the following problem: Let $(X_k, Y_k)_k$ be a sequence of $2$-dimensional, independent random variables, each with uniform distribution over $ B(0,k) $ Verify if the following ...
0
votes
0answers
13 views

Controlling a random variable

I've got a system the output of which is a random variable with a certain distribution, which for the purpose of this discussion can be assumed to be normal. The input variable is voltage. The ...
1
vote
0answers
23 views

Proving asymptotic normality

Suppose I have a sequence of independent random variables. Then how do I prove $S_n\sim\mathrm{AN}(0,n)$. Lyapunov or Lindeberg's CLT not working since $V(S_n) \neq n$ and the Characteristic function ...
0
votes
1answer
35 views

Probability, need help on two Random Variable

I have a question about computing the probability of two random variable. Let $X$ and $Y$ be two independent random variable with distribution $f_x$ and $f_y$ such that they are only define for ...
3
votes
0answers
32 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that ...
1
vote
0answers
52 views

Sum of poisson random variables

Let $N, X_1, \dots , X_n$ be independent random variables. $N \sim P(\lambda) \quad (\text{Poisson distribution})$, while $X_k \sim B(p)$ (Bernoulli) Let us consider the "random" sum $S = X_1 + ...
1
vote
1answer
34 views

Rescaling function for probability of $k$ adjacent ones in a binary string

Call $\xi$ a random variable taking values in $\{0, 1\}^{\{0, 1, 2, \ldots, n\}}$, where each character of the string has vaalue $1$ with probability $p$ and $0$ with probability $1-p$ independently. ...
0
votes
1answer
39 views

Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
4
votes
1answer
58 views

What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
0
votes
2answers
52 views

Random variable distribution. Reposted

$X$ has distribution $B (30, 0.6)$. Find $P(X \geq 16)$. I know how to find $2$ or $3$ numbers where you use combinations and simply add probabilities for each variable. But this value includes $14$ ...
-2
votes
1answer
32 views

What is the pdf of the area of a rectangle having sides $X$ and $2-X$, with X r.v. in $(0,2)$? [closed]

Let be $X$ a random point chosen in the interval $(0,2)$. Find the pdf of the area of the rectangle having sides $X$ and $2-X$.
1
vote
2answers
44 views

Problem on Convergence of random series

Suppose that $\{X_n\}$ is an independent sequence and $E[X_n]=0$. If $\sum \operatorname{Var}[X_n] < \infty$, then $\sum X_n$ converges with probability $1$. Is independence necessary condition ...
2
votes
0answers
37 views

Poisson to Binomial Distribution Proof?

Q:Let {N(t) : t ≥ 0} be a Poisson process. For s = t/3, show that the conditional distribution of N(s) given N(t) = n is binomial with parameters n and p = 1/3. Also, find the conditional distribution ...
2
votes
1answer
17 views

Probability that Modulus of Difference Greater than $3$

Given that: $X$ and $Y$ be two Continuous Random Variables with Joint pdf $$f_{XY}(x,y)=\frac{1}{x^2y^2}, x \ge 1\,\,,y \ge 1$$ Find $$P\left(|Y-X|\gt 3\right)$$ My Try: Its Clear that $X$ and $Y$ ...
0
votes
2answers
55 views

Pdf of $Z=(XY)^{1/2}$. with X,Y independent r.v. with the same distribution (iid) [closed]

Let be $X,Y$ two independent random variables having the same distribution (the following is the density of this distribution) $$f(t)= \frac{1}{t^2} \,\,\, \text{for $t>1$}$$ Calculate the ...
2
votes
2answers
46 views

What's the density of $Z=\max(X,Y)-\min(X,Y)$ with $X,Y$ exponentials of parameter $\lambda$?

Let be $X,Y$ two independent exponential random variables with parameter $\lambda$. What is the pdf of $Z=\max(X,Y)-\min(X,Y)$? Thanks for your help.
0
votes
0answers
17 views

Auto Covariance of Square Law Detector

if $X(t)$ is Zero Mean WSS Gaussian Random Process , Find the Auto Covariance of $$Y(t)=X^2(t)$$ we have $$E(Y(t))=E(X^2(t))=E(X^2(t+\tau))=R_X(0)$$ $$C_Y(\tau)=R_Y(\tau)-R_X^2(0)$$ ...
1
vote
0answers
21 views

Size-bias coupling, poisson approximation, telescoping sum

Good day I was reading this lecture. I don't understand the proof of theorem $4.13$, which is on page $252$. Theorem 4.13: Let $W \ge 0$ be an integer-valued random variable such that ...
3
votes
3answers
112 views

Laws of large numbers and independence

I just did a brief review of various sources, and they all specify that if $X_i$'s are independent, identically distributed random variables, then $S_n/n \rightarrow E(X_i)$ (with respect to various ...
-1
votes
1answer
52 views

What's the pdf of $Z=X^2 +2X$ if $X$ is a standard normal? [closed]

Le be $X$ distributed as a standard normal. What is the density function of $Z=X^2 +2X$? Thanks for your help
1
vote
3answers
62 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
1
vote
2answers
32 views

Inequality between random variables

Let $X_k$ be $\text{i.i.d.}$ continuous random variables. Find in terms of $n:$ $$\mathbb{P}\Big(X_1\geq X_2\geq\cdots \geq X_{n-1}<X_n\Big)$$ Let each $X_k$ have $\text{p.d.f }\;f$ and ...
2
votes
3answers
77 views

How can expected value be infinite?

My book as well as Wikipedia gives this definition of expected value: $\mathbb E(X)=\sum _x xf(x).$ But, $\mathbb E(X)$ is said to exist if and only if that equation is absolute convergent. ...
0
votes
1answer
39 views

continuous RV from discrete RV

So I am reading some notes in stochastic processes and I don't really understand the solution of this problem: Problem: Let $(\Omega,F,\mathbb{P})$ be a probability space where $\Omega$ is the set ...
2
votes
1answer
38 views

Removing a fixed quantity from multiple “buckets” randomly

Suppose I have a set of $100$ elements split into $4$ buckets A-D as follows: A: 10 elements B: 20 elements C: 30 elements D: 40 elements I want to remove $k ...
0
votes
3answers
39 views

Independent variables, normal distribution, pdf

I have independent variables $ X_1, X_2,\ldots,X_n $ with normal distribution on range $ [0,1] $ . Next, variables $ Z_i $ are created according to this formula $ Z_i = - \frac{1}{\lambda} \ln(1-X_i) ...
0
votes
0answers
20 views

Prove that random variable has exponential distribution

I have a non-negative continuous random variable $X$ and I am given $P(x>a+b)=P(x>a)P(x>b)$. I need to prove that the probability distribution is exponential. I already know that $F(x)=0$ if ...