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

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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 ...
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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
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2answers
51 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 ...
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1answer
12 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 ...
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0answers
24 views

Question about notation for a statement about conditional probability distribution

Consider the random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, P)$. Suppose $Y$ is a discrete random variable with support $\mathcal{Y}\subset \mathbb{R}$. Suppose ...
0
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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 ...
2
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1answer
28 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
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3answers
67 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
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0answers
47 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
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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 ...
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1answer
13 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
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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 ...
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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
53 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
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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 ...
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0answers
29 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 ...
1
vote
2answers
29 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 ...
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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
39 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 ...
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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
73 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
87 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
42 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
33 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
55 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 ...
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0answers
23 views

Probability Density Function of Random Process

I have a signal s(t)=t for t is between 0 and 3. I have a uniform random variable A between 0 and 10. The random process is defined as s(t-A). What is the probability density function of the process?
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0answers
17 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
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0answers
30 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
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1answer
18 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
72 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 ...
0
votes
1answer
28 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 ...
1
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0answers
26 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
57 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 ...
0
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1answer
36 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
50 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 $P(X<Y)\neq 1/2$ ...
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votes
1answer
16 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) = ...
6
votes
1answer
87 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
24 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, $X_1$, $X_2$, ...
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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 ...
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1answer
20 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 ...
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0answers
18 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, ...
1
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2answers
32 views

the definition of random variable

If we supposed that X is a random variable, is X - X a random variable? Could the outcome of an event is only 1? Cause X-X has only one outcome, and the possibility of it is 1. How about X + X?
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0answers
17 views

Two different random processes for generating polynomials

Consider two processes for generating random complex polynomials: choosing the roots uniformly and independently throughout the unit disc, and choosing the coefficients uniformly and independently ...
1
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0answers
37 views

Mean Value of a Random Process

Consider a random process $X(t) = Z(t)\sin(wt-Q)$. Here $Q$ is a random variable taking values $q$ in $[-\pi/2,\pi/2]$ with PDF given by $$p_1^Q(q) = \frac{\cos(q)}{2}$$ $Z(t)$ is some random ...
0
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0answers
29 views

What is the joint distribution of order statistics and samples?

If samples $X_1, X_2, ... X_t$ are picked independently and identically from the uniform distribution $[1,2, ..., P]$, what is the joint distribution ...
1
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0answers
53 views

average of reciprocal of sum of random variables

I have a slightly more complex problem, but I believe the technical nature is captured in the following: For the $s$-indexed sequence, $X_s$, of iid positive-valued random variables with finite ...
0
votes
1answer
50 views

Central Limit Theorem; Exponential Distribution

I'm trying to prove the Central Limit Theorem for the exponential distribution and I'm running into problems. This is what I've done so far: Given $S = X_1+X_2+...+X_n$ where each $X_i$ is an ...