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

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0
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1answer
35 views

Finding PDF of function of a random variable

Suppose $X$ has PDF: $f_X (x)= \lambda e^{-\lambda(x+2)}$ , for $x \ge-2$ $f_X(x)=0$ , for $x <-2$ Determine the PDF of $Y = X^2$. I am stuck because for $-2\le X \le 2$, $0\le Y \le 4$, and I ...
3
votes
0answers
52 views

$P(|X_1+X_2|<x)\le P(|X_1|<x)$ for every independent centered continuous $X_1$ and $X_2$?

Let $X_1$ and $X_2$ be zero mean independent continuous random variables. Then, is it true that $P(|X_1+X_2|<x)\le P(|X_1|<x)$. The intuition is that summing independent variables increase ...
1
vote
0answers
17 views

Multivariate normal distribution problem

Consider three Gaussian variables $X_1,X_2,X_3$ with $\mathbb{E}[X_i]=0$ and $\mathbb{E}[X_iX_j]=\rho_{ij}$ for $i,j=1,2,3$. Then, three new variables are defined: $$ \left\{ \begin{array}{l1} Y_1 ...
0
votes
1answer
33 views

What is $E[Z|Z\ge 0]$ where $Z$ is a continuous random variable with support in $[-1,1]$?

I have a random variable $Z$,I seek an expression for $E[Z|Z \geq 0]$. I assume this is easy to get hold of but I just can't seem to get it. As a further complication $Z=X-Y$, where $X$ and $Y$ are ...
5
votes
1answer
75 views

On the linear combination of $\pm 1$ random variables

Let $X_1,\dots, X_n$ be i.i.d symmetric $\pm 1$ random variables, i.e. $X_j$ takes values in $\{-1,1\}$ with $$\mathbb{P}(X_j=1)=\mathbb{P}(X_j=-1)=\frac{1}{2}.$$ Let $a_1,\dots,a_n\in\mathbb{Z}$, ...
-2
votes
2answers
33 views

Generating a random variable from a uniform random variable [closed]

I have no idea how to go about doing this. Any help would be much appreciated.
2
votes
1answer
61 views

How many students would have to take the exam to ensure with probability at least $.9$ that the class average would be within $5$ of $75$?

I'm having trouble solving this problem: From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean $75$. How many ...
0
votes
2answers
43 views

Find the density function from a joint density function

I try to solve the following task and I don't know what the correct way to do is. Let $p\in(0,1)$ and $(X,Y)$ be a pair of random variables with distribution density function $$f(x,y)=\frac{1}{2\...
0
votes
2answers
24 views

Two discrete r.v. problem, joint density

Problem A cook needs two cans of tomatoes to make a sauce. In his cupboard he has $6$ cans: $2$ cans of tomatoes, $3$ of peas and $1$ of beans. Suppose that the cans are without the labels, so he can'...
0
votes
0answers
6 views

Finding the norm of estimation error asymptotically

Let $\theta \in \mathbb{R}^p$ be such that it has uniform distribution on the set of standard unit vectors $\{\tau e_1,\ldots,\tau e_p\}$, for $\tau=\sqrt{(2-\varepsilon)\log p}, \varepsilon>0$. ...
0
votes
2answers
61 views

Probability problem with combination of poisson and binomial distributions

Exercise The number of clients that enter to a bank is a Poisson process of parameter $\lambda>0$ persons per hour. Each client has probability $p$ of being a man and $1-p$ of being a woman. After ...
0
votes
1answer
15 views

Limes superior and random variables

I want to show the following: Let $X_1,X_2\dots$ be i.i.d. random variables. Let $\text{E}[|X|^p]=\infty$ for $p>0$. Show that $$P(\limsup\limits_{n\to\infty }\{|X_n|\geq n^{1/p}\})=1$$ What I ...
0
votes
1answer
10 views

Extending Random Number Ranges

I am provided with a random number $\xi \in [0,1]$. I check if a particular $\xi_i \lt x$ is true and if so, I need to convert those random numbers within the range $[0,x)$ into uniform range in $[0,1]...
2
votes
1answer
31 views

probability problem with Poisson distribution

Problem A retailer knows that the demand of boxes is a random variable with Poisson distribution of parameter $\lambda=2$ boxes per week. The retailer completes his stock on monday so as to have four ...
2
votes
3answers
71 views

Picking two random points on a disk

I try to solve the following: Pick two arbitrary points $M$ and $N$ independently on a disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2 \leq 1\}$ that is unformily inside. Let $P$ be the distance between those ...
0
votes
0answers
49 views

Sequence of non-independent coin tosses

Suppose that a sequence of coin tosses is due to be performed. Let $p_i$ denote the probability that the $i$th coin toss lands on Heads and let $X_i$ denote the corresponding indicator random variable ...
2
votes
0answers
21 views

Almost sure convergence and limes superior

I'm trying to prove the following exercises and I don't know if my attempts are correct. A sequence of real random variables $(X_n)$ almost surely converges to $X$ if and only if for every $\epsilon ...
0
votes
1answer
14 views

Continuous and discrete random variables defined on the same probability space?

I am confused on the definition of continuous/discrete random variables defined on the same probability space. Consider the random variables $X,Y$ defined on the same probability space $(\Omega, \...
0
votes
1answer
17 views

Probability problem with random vectors

Problem Suppose that $10$% of the american population smokes dark cigarettes, $35$% smokes white cigarettes, $3$% smokes pipe and the rest of the population doesn't smoke. A group of $35$ persons was ...
0
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0answers
28 views

Probability with binomial distribution and random vectors

In a city the proportion of men with blue eyes is $20$%, of green eyes is $5$%, of black eyes is $10$% and the rest $65$% of men has brown eyes. Susan decides to commute from the center of the city to ...
0
votes
1answer
36 views

expected value of fisher distribution

I know that the pdf of an F-distribution is $f_{k,m}(t) = \Gamma(t)=\frac{\Gamma((k+m)/2)}{\Gamma (k/2)\Gamma(m/2)}k^{k/2}m^{m/2}t^{k/2 - 1}(m+kt)^{-(k+m)/2}$ Also, $E(F)=\int xf_{k,m}dx$. How do ...
0
votes
3answers
54 views

Uncorrelating random variables.

I was reading this answer, and the first sentence seemed more intuitive at first than after thinking through it: If $\pmatrix{X\\ Y}$ is bivariate normal with mean $\pmatrix{0\\0}$ and ...
0
votes
0answers
21 views

symmetry of two IID random variables [duplicate]

Suppose that $X$ and $Y$ are independent and identically distributed. The claim is that $P(X<Y)=P(X>Y)=1/2$. How do I prove this? My attempt Since they are IID $f_X=f_Y$. So $P(X<Y)=\int\...
1
vote
0answers
31 views

Upper bound for $\frac{\|x\|_1}{\|x\|_2}$ if each entry of $x\in R^d$ is i.i.d. sampled from Gaussian distribution $N(0,1)$

In the question, $\|x\|_1=\sum_{i=1}^d|x_i|$ with $|\cdot|$ being the absolute value, and $\|x\|_2=\sqrt{\sum_{i=1}^d x_i^2}$. In general, $\frac{\|x\|_1}{\|x\|_2}\leq \sqrt{d}$ always holds for ...
2
votes
2answers
36 views

Expectation and variance of matrix valued random variable

Suppose I have a discrete matrix-valued random variable $X$, that is, I have defined a set of fixed matrices $\{Y_i\}_{i=1}^n$, and the random variable $X = Y_i$ with probability $\frac{1}{n}$. Is ...
-2
votes
1answer
21 views

Expectation of a jointly distributed Random Variables, does it exist?

I see this equation in a paper published in a proceeding of a very competitive conference:( and thus I don't think it has a flaw) $$Pr(Y|X)= \frac{e^{-E(X,Y)}}{g(X)};$$ Where $X,Y$ are two random ...
0
votes
1answer
41 views

Intersection of infinite number of compact sets

Let $\mathbf{y}_k = \mathbf{y} + \mathbf{e}_k$, where, $k \in \mathbb{N}$, $\mathbf{y}, \mathbf{e}_k \in \mathbb{R}^n$, $\mathbf{e}_k$ is a sequence of i.i.d. random variables, and $E \subset \mathbb{...
1
vote
0answers
34 views

Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0 < P(B) < 1$

Let $(\Omega, \mathbb{F}, P)$ be a probability space. Prove that $I_A$ is a random variable with $A$ being the union of events $B$ $\in$ $\mathbb{F}$ such that $0$ $<$ $P(B)$ $<$ $1$. My ...
0
votes
1answer
22 views

Probability for random vector given probability distribution [closed]

Given the following probability distribution: $f(x,y) = \begin{cases} xe^{-x-y}, & x,y>0 \\[2ex] 0, & \text{elsewhere} \end{cases}$ compute $P(X \le Y)$. I know that the result is $1/4$, ...
1
vote
1answer
74 views

Random Variables in a Uniform Probability Space

Suppose that $\Omega = \{1,2,3,4,5,6\}$ is a uniform probability space. Now, let $X(\omega)$ and $Y(\omega)$, for $\omega \in \Omega$, be random variables defined as: $$\begin{array}{|c|c:6c|} \...
2
votes
1answer
25 views

Expectation of the fraction a random function covers its range

Preamble: The number of onto functions from a set of $m$ elements to a set of $n$ elements is, as stated in this answer, computed as follows: $$n!{m\brace n}\;.$$ Now, let's count the number of ...
1
vote
2answers
93 views

Expected number of women sitting next to at least one man?

There are $10$ seats, $5$ men and $5$ women who randomly occupy these seats. I have to calculate the expected number of women sitting next to at least one man. My attempt: I defined a random variable ...
2
votes
1answer
47 views

Borel-Cantelli exercise

I'm stucked with this exercise. Let $X_1,X_2,\ldots$ be i.i.d. random variables with $E(X_1)=0$ and $Var(X_1)=\infty$ Prove that$$P(\limsup\limits_{n\to\infty}\{|X_n|\geq \sqrt{n}\})=1$$ I need to ...
1
vote
1answer
34 views

Convergence of series of random variable without distribution

I'm trying to solve the following task and I'm struggling very much. I don't know if it is correct what I did so far. Let $(X_n,n\geq1)$ be a sequence of independent random variables such that $\...
0
votes
2answers
58 views

expectation of the number of empty cells

You are given a random number, $N$, of balls, where $N$ has a Poisson distribution with parameter $\lambda > 0$. You then place these balls one by one among $r$ ($\geq 2$) cells according to the ...
0
votes
0answers
21 views

The quotient of two chi distributions

The quotient distribution of two chi-squared distributions is F-distribution. What would be the quotient distribution of two chi distributions? Is there a general distribution for this?
2
votes
0answers
36 views

Limiting distribution of infinite sum of weighted bernoulli?

Let $p_n$ be some fixed pulse, for example $p_n =e^{-n^{2}}$ We have an infinite sum $y = \sum_{n=-\infty}^{\infty} a_n p_{-n}$ where $a_n$ are iid bernoulli random variables taking the values $+/- \...
1
vote
1answer
20 views

Sum of two random variables converging with different modes [closed]

Is it true that if X_n converges in distribution to X; Y_n converges in probability to Y; X_n, Y_n, X and Y are real-valued random variables defined on the same probability space, then X_n + Y_n ...
3
votes
1answer
74 views

Median of a multinomial variable

Let $k\in\mathbb N^+$ be a positive integer. Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$. For $i\in \{1,2,\...
0
votes
1answer
29 views

memoryless property of exponential distributions with random variables

It is true that $P(X>t+s|X>t)=P(X>s)$ for certain values $t$ and $s$. However, how can I show that this still holds if: $T$ is a continuous random variable. That is $P(X>T+s|X>T)=P(X&...
1
vote
0answers
28 views

Gap probability for i.i.d. random variables

Given a set $\{X_1,\ldots,X_N\}$ of real i.i.d. random variables, drawn from a common parent pdf $p_X(x)$, what is the probability that, given one random variable taking value in $(t-dt,t)$, there are ...
2
votes
2answers
59 views

Showing two random variables independent despite seemingly looking dependent

I just met this in probability and it got me completely stumped: We define an i.i.d sequence of normally distributed random variables $ \{ X_n \}_{n=1}^{\infty} $ such that $ X_n \sim \mathcal{N}...
0
votes
1answer
40 views

Simple questions about random variable

1. Poisson random variable takes infinite number of values ? 2. A binomial random variable takes infinite number of values ? I guess both sentences are true. Namely, I think both of random variable ...
0
votes
1answer
52 views

Probability of two IID random variables

Let $X$ and $Y$ be independent and identically distributed. Show that if $X$ and $Y$ are continous, then $P(X<Y) = 1/2$ Give an example of two IID RVs $X$ and $Y$ such that $P(X<Y)\neq 1/2$ ...
0
votes
1answer
21 views

Calculating Entropy and Information Gain of a Variable

I have the following values for two random variables. I need to compute the following values: a. H(Y) b. H(Y|X) c. and finally IG(Y|X) I will show what I have calculated so far. a. H(Y) = -(.5*...
6
votes
1answer
91 views

A pill bottle with large and small pills

Alright here's the exact question: A bottle initially contains $48$ large pills and $76$ small pills. Each day a patient randomly chooses one of the pills. If a small pill is chosen, it is eaten. If ...
0
votes
1answer
26 views

What is the domain of this random variable?

I've been self-studying Introduction to Statistical Learning. From page 16 of the book: "...suppose that we observe a quantitative response $Y$ and $p$ different predictors, $X_1$, $X_2$, $\ldots$,...
0
votes
0answers
21 views

what is the relation between X and ω

From the definition of random variable: In the special case of probability space (Ω, F, P), we use the phrase random variable (RV) to mean a measurable function, that is, X : Ω → R is a random ...
0
votes
1answer
22 views

Calculate expectation of Cumulative distribution function of a normal distribution

I have to calculate the expectation of the Cumulative Distribution Function of a normally distributed random variable X, which has variance 1 and mean 0. I calculated the integral of the CDF (taken as ...
1
vote
0answers
19 views

Expectation of the maximum of random variables

I'm trying to get $E(\max \{ a-X, b-X-Y, 0 \})$, where $X$ ~ $N(0,\sigma^2)$, and $Y$ ~ $N(\mu, \gamma^2)$, and $X,Y$ are independent. I've been trying to figure this out by doing, $E(\max \{ a-X, ...