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

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2answers
45 views

How to compute the following covariance?

$\alpha(t),\beta(t)$ are two stochastic process. How to prove the following equation: ...
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1answer
28 views

Joint normal random variables, covariance, and probability

I'm having a lot of trouble with this question: X and Y are joint normal random variables with common mean 0, common variance 1, and covariance 1/2. What is $P(X+Y\leq \sqrt{3})$? Thank you!
1
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1answer
32 views

Sample Space of a Product of Random Variables

Given a fair coin with $1$ on one side and $0$ on the other, $X$ is defined as "result of 1st coin flip" and $Y$ is "sum of results of both flips". $Z$ is defined as "$X·Y$". For example, $X=1 = ...
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0answers
37 views

Random sum a.s convergence and convergence in probability

Let $X_n$ be a sequence of independent random variables such that $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Consider the sum $S_n=\Sigma_{k=1}^nX_k.$ How can we show that for any ...
0
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1answer
80 views

Uniformly random subset

Two people visit their favorite pub that has 10 diff erent beers. They both order, independently of each other, a uniformly random subset of 5 beers. One of the beers available is Leo's Early ...
0
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1answer
103 views

Fair coin flipped twice independently

Consider a fair coin that has 0 on one side and 1 on the other side. We flip this coin, independently, twice. Define the following random variables: X = the result of the first coin flip Y = the sum ...
0
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1answer
31 views

Expected value of multiple of 2 random variables

I have to random variables. X with the values -1, 0, 1 each with the probability 1/3, and Y with the value 0, 1 with the probability 1/3 and 2/3 respectively. As part of the questions have to find ...
1
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1answer
33 views

Calculating the variance of an i.i.d variable

Here is the data I have for a variable: X = [420 450 420 380 440 380 360 360] I'm told that X is i.i.d with mean = 390. How do I go about estimating variance? Would it simply be: (Σ(x – xbar) ^2 ...
2
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1answer
46 views

If $N$ is an integral random variable, can $S_N$ be expressed in terms of the $S_n$ and $P(N=n)$?

Suppose $X_1,X_2,...$ is an infinite sequence of i.i.d random variables and let $N$ be a positive integer valued r.v independent from $\{ X_i \} $. Let $S_n = \sum_{i=1}^n X_i $ and $S_N = X_1 + ... + ...
0
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3answers
101 views

Consider two random variables X and Y

Consider two random variables $X$ and $Y$. If X and Y are independent random variables, then it can be shown that: $$E(XY) = E(X)E(Y).$$ Let $X$ be the random variable that takes each of the values ...
3
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1answer
59 views

Series of independent random variables are independent again

Let $\{X^i_j : i=1,..n, j\in\mathbb{N}\}$ be an independent set of random variables on a probability space $(\Omega, A, \mathbb{P})$, $$X^k_l: \Omega \to \mathbb{R^+} := \{x \in \mathbb{R} : x \ge ...
1
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1answer
114 views

Expected value and indicator random variable

During a period of $n$ days, two persons drink beers, one each. There are $n$ different beers $B_1, B_2, B_3,\ldots, B_n$, where $n\geqslant 1$ is an integer. t Person1 drinks the beers in order, ...
1
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1answer
91 views

The probability that the ratio of two independent standard normal variables is less than $1$

Let the independent random variables $X,Y\sim N(0,1)$. Prove that $P(X/Y < 1) = 3/4. $ Could anyone help me prove this analytically? Thanks. Progress: My first thought was to integrate the joint ...
0
votes
1answer
36 views

Conditional Expectation as a random variable of independent rendom variables

Given two independent random variables $X_1$, $X_2$ and $X_3$, then $Y=F(X_1, X_2, X_3)$ is a random variable depending on $X_1$, $X_2$ and $X_3$. Would some one help me to detect whether the two ...
2
votes
2answers
85 views

Expected value of random permutation

Let n ≥ 1 be an integer and consider a uniformly random permutation $a_1$, $a_2$, . . . ,$a_n$ of the set {1, 2, . . . , n}. Define the random variable X to be the number of indices i for which 1 ≤ i ...
0
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1answer
27 views

Consider a communication source that transmits packets containing digitized speech…

Consider a communication source that transmits packets containing digitized speech. After each transmission, the receiver sends a message indicating whether the transmission was successful or ...
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0answers
52 views

multiplying a random variable by another random variable?

Consider a fair coin that has 0 on one side and 1 on the other side. We flip this coin, independently, twice. And we have the following random variables: ...
0
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0answers
57 views

Cat and mouse 'time until eaten' problem

I'll start by saying that this is for a homework, so any hints will be more appreciated than straight up answers, as I would like to obtain the answers through my own working. Anyway, I'll just copy ...
2
votes
2answers
49 views

Inequality for a random variable

Let $\xi > 0$ be a random variable such that $\mathbb E[\exp{(-\xi)}]=\exp{(-a)}$. How to show that for any $c>0$ $$P(\xi \ge c) \le \frac{1-\exp{(-a)}}{1-\exp{(-c)}}$$
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0answers
19 views

matrix with random components

Is it possible to have a matrix some components being random variables? (Random variables in the statistical sense, mapping from $\Omega$ to $\mathbb{R}$.) If so, is it possible that the eigenvalues ...
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1answer
28 views

Expectation through survival function

I understand how we get $$E[x] = \int_0^\infty (1-F(x)) dx$$ but why does this apply only to non-negative random variables
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1answer
32 views

Distribution of the lifetime of a system consisting of two exponentially distributed components, one being backup

I have a system consisting of components $S_1$ and $S_2$ whose lifetimes $T_1$ and $T_2$ follow the exponential distribution with parameter $\lambda$. At time $t=0$ the component $S_1$ is switched on ...
4
votes
1answer
265 views

Uncorrelated but not independent random variables

Is it possible to construct two random variables $X, Y$ both of them assuming exactly two non-zero values which are uncorrelated, i. e. $\mathbf{E}[X \, Y] = \mathbf{E}[X]\,\mathbf{E}[Y]$, but not ...
1
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2answers
31 views

mapping random variables

Let $x$ be a random variable (RV), $a<x<b$ with a pdf $f(x)$. Let's construct a function on $x$, $y=h(x)$ which is continuous and differentiable. If an inverse function exists for $h$, say ...
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0answers
24 views

How to find the cumulative distribution function of a sum of n continuous random variables?

Consider the sequence $\{t_n\}_{n\ge 0}$ of independent and identically distributed continuous random variables with distribution function $F(x)$, i.e., $P(t_n\le x)$. Define $~~T_n=\begin{cases} ...
2
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1answer
80 views

Can someone explain Wishart distribution?

I have to use Wishart distribution to model noise in images. Can someone explain or give intuition behind wishart distribution. Thank you !!!
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1answer
54 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
0
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1answer
41 views

Check/Prove if random variable

I am having some doubts on how to prove/check if something is a random variable. The question I am trying to solve currently is: 1) Consider X a random variable on space (Ω,F,P). If [X] is the ...
1
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0answers
60 views

Bounds on dependent random variables

Consider this problem: You roll a dice n times. Let $ X_i $ be the number of times you see face $i$. Specifically, I am interested in $ E[ \max\limits_{i} X_i] $. In fact, I don't even need an exact ...
1
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1answer
26 views

Estimation of the Probability of $\left\{|X^{(i)}-X^{(n)}|\le2\delta\text{ for all }i\in[n,m-1]\text{ and }|X^{(m)}-X^{(n)}|>2\delta\right\}$

Assumptions Let $(X_i)_{i\in\mathbb{N}}$ a sequence of independent real-valued random variables and $$B_{m,n}:=\left\{\left|X^{(i)}-X^{(n)}\right|\le2\delta \text{ for all }i\in[n,m-1]\text{ and ...
1
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1answer
45 views

How to argue independence of random variables

I'm having trouble with a pretty basic idea- I just don't know how to argue that two events are independent. I know the definition: $$P(A)P(B)=P(A\cap B)$$ but if we are given two events, like the ...
2
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1answer
41 views

Bounds on $E[(f(X)-g(X))^2]$

I am looking for good upper bounds on $E[(f(X)-g(X))^2]$. For example, here are two bound that I derived: $$ E[(f(X)-g(X))^2] \le E[4 \max(f(X)^2, g(X)^2)] $$ where I used $(x-y)^2 \le 4 ...
1
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0answers
21 views

Finding the conditional distribution of a normal RV given another normal RV

I'm having trouble with this question from a past qualifying exam: Question Suppose $Z \sim N(\mu,\sigma^{2})$, $W \sim N(0,1)$ and $V \sim N(0,1)$ are mutually independent normal random variables. ...
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0answers
16 views

What is $\operatorname{Var}(x^y) x$ is normally distributed random variable?

I am trying to understand the $\operatorname{Var}(x^y)$. I thought that it is $E(x^2)^y-[E^2(x)]^n$ but I understand that I was wrong, or wasn't I?
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4answers
96 views

Conditional distribution of order statistics

Let $X_{(1)},...,X_{(n)}$ be the order statistics of a set of $n$ independent uniform $(0,1)$ random variables. Find the conditional distribution of $X_{(n)}$ given that ...
1
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2answers
35 views

Change of Variables in Second Moment

$X$ is a non-negative continuous random variable with pdf $f(x)$. $G(t) = \int_{t}^\infty f(x) dx$. Show that $$E[X^2] = 2 \int_{0}^\infty tG(t) dt$$. I tried to write out E[X^2] (second moment of ...
1
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1answer
31 views

conditional probability has binomial distribution

Let $X_1, X_2$ be two independent random variables with $X_i \sim \mathrm{Pois}(\lambda)\,$ for $i=1,2$, where $\lambda>0$. Let $k,n \in \mathbb{N}$ and $0\leq k \leq n$. Define ...
1
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1answer
31 views

Existence of a random variable given a cdf

For every real function F which can be a CDF (so has the properties that $F(+\infty)=1$, $F(-\infty)=0$, and F is non-decreasing and right continuous), does there exist a random variable on a ...
0
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1answer
33 views

Expectation of a function of a normal random variable

Suppose $X\sim\mathcal{N}(0,1)$. I would like to find $\mathbb{E}[\frac{1}{\alpha+\beta X}|A<X<B]$ where $A, \alpha, \beta>0$. How should I go about it? Finally, if the answer is that there ...
5
votes
2answers
33 views

correlation between $\sum_{i=1}^{98}X_i$ and $\sum_{i=3}^{100}X_i$

Let $X_1,...,X_{100}$ be iid $N(0,1)$ random variables. The correlation between $\sum\limits_{i=1}^{98}X_i$ and $\sum\limits_{i=3}^{100}X_i$ is equal to (A) $0$ (B) $\dfrac{96}{98}$ (C) ...
0
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0answers
23 views

Order statistics difficult problem

The $n+1$ random variables $X_i$ ($1\le i\le n+1$) are independent and identically distributed with cummulative distribution $F$. Let $Y_k$the order statistics of $X_1,...,X_n$ and let $Z_k$ the order ...
1
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1answer
23 views

Exponential order statistics

Let $X_1,...,X_n$ exponential random variables with parameter $\lambda$ and let $X_{(1)},...,X_{(n)}$ the order statistics of the random variables. I know that $X_{(1)}$ is exponential with parameter ...
2
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1answer
20 views

An independent sequence of square-integrable random variables with convergent sum of variances converges stochastically

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of independent and square-integrable random variables with $\operatorname{E}\left[X_i\right]=0$ and ...
0
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0answers
69 views

Expected value of a function of truncated normal

I need to find the expected value of the following type of an expression: $$\mathbb{E}[\frac{1-\alpha}{1-\alpha-\frac{X}{\beta}}]$$ where $\alpha$ and $\beta$ are constants and $X$ is a random ...
2
votes
3answers
39 views

Let $X$ ~ $Exp(1)$, and $Y$ ~ $Exp(2)$ be independent random variables. Let $Z = max(X, Y)$. calculate $E(Z)$

Here's a question I'm trying to solve: Let $X$ ~ $Exp(1)$, and $Y$ ~ $Exp(2)$ be independent random variables. Let $Z = max(X, Y)$. calculate $E(Z)$ I'm can't understand how to deal with ...
0
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2answers
27 views

If $X_1,X_2…$ are independent, uniform random variables do there exist an infinite amount of $Y_n = X_nX_{n+1} < \frac{1}{8n}$?

If $X_1,X_2...$ are independent, uniform random variables on the interval $[0,1]$, do there exist an infinite amount of $Y_n = X_nX_{n+1} < \frac{1}{8n}$? I want to use the Borel-Cantelli lemma, ...
1
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0answers
29 views

Conditional probability in bivariate distribution

Suppose I have a multivariate normal distribution $N$ for the continuous multivariate RV $X = (X_1, X_2)^T$. It is clear that it has to hold $\int N = 1$ i.e. probabilities sum up to one. Suppose I ...
1
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2answers
53 views

Distribution of a random measure is determined by the characteristic function

I ham trying to understand a proof from a book I am reading. It says the proof follows directly from the prior theorem and I just can't see that. Let $X$ be a random measure on a locally finite, ...
2
votes
0answers
82 views

Tail bound on the sum of independent identical geometric random variables

Suppose $X_1,\ldots,X_k$ are k independent geometric random variables with the same success probability and let $X=X_1+\cdots+X_k$. Hence $E[X_i] = 1/p$, the expected number of trials needed is ...
2
votes
1answer
52 views

Expected value complex random variable

I want to check that if $X: \Omega \to \mathbb{C}$ is a random variable, then the inequality $| \mathbb{E} X| \le \mathbb{E} |X|$ also holds like in the real case. We can write $$X = \Re X + i \cdot ...