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

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0
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3 views

Expected Value of $P(|Y_n^{(K)}| > \epsilon)$ where $Y_n^{(K)}$ is the random sum of a sequence of RV converging to 0 in Probability

I have been struggling with this for countless hours, I would appreciate a hint to get me going in the right direction (no complete answer please) Problem: Assume that for all $k \in \mathbb{N}$ ...
1
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1answer
9 views

Probability of intersection of multiple 2-way universal events

If given a set of events that are known to be 2-way universal, is there a closed-form solution for the probability of their intersection? If so, how would you go about finding it? I know that for 2 ...
1
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1answer
23 views

Proving that $|P\{X=m\}-P\{Y=m\}| \leq P\{X\neq Y\}$

Let $X$, $Y$ be random variables on the same probability space. Show that for all $m$, $$|P\{X=m\}-P\{Y=m\}| \leq P\{X\neq Y\}$$ I'm actually not even sure how to start. I think it's going to ...
0
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0answers
6 views

representative sample

population is students taking a Chemistry class; Sample of 60 students as a random sample from this population. Select one variable on the survey and argue whether or not you think this sample is ...
0
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1answer
9 views

Difference between random number and random variant?

After generating random number we can get the random variant by using inverse transform or other techniques. What is the difference between random number and random variant. Can anyone explain it with ...
0
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1answer
32 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
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0answers
43 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 ...
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0answers
16 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} ...
0
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1answer
31 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 ...
0
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0answers
30 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 $a_1,\dots,a_n\in\mathbb{R}$, ...
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2answers
28 views

Generating a random variable from a uniform random variable [on hold]

I have no idea how to go about doing this. Any help would be much appreciated.
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1answer
36 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
36 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 ...
0
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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
votes
2answers
41 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
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1answer
11 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
21 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
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 ...
2
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1answer
25 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
62 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 ...
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0answers
28 views

how find an event with probability $6.6p^2q$ [on hold]

How to find an event with the following probabilities $6p^2q$,$6.6p^2q$,$6.75p^2q$,$3.9pq$,$4pq$ using independent bernoulli trials?
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0answers
44 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
19 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
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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
votes
1answer
14 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
35 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 ...
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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
27 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
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2answers
25 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 ...
-4
votes
0answers
53 views

Affine function of cdf [closed]

Suppose a random variable X has cdf FX(·). Express the cdf of the following random variables: X + b aX + b |X| max(X,0) Could someone show me how to use the given random variable X and its ...
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 ...
1
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0answers
33 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
21 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 $1/4$, ...
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0answers
32 views

Derivation of pdf from the function of random variables [closed]

Let $A_{i}$ and $B_{i}$ ($i=1,...,K$) be the random variables of which pdf/cdf are known to us. And, there is a function of random variables, $C=\max({A_{1}+B_{1}, A_{2}+B_{2}, ..., A_{K}+B_{K}})$. ...
1
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1answer
70 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
24 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
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2answers
83 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
35 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
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1answer
32 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
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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
22 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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16 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
27 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
71 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
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0answers
27 views

Probability density function with two peaks and skewness

I have plotted a probability density function on a graph. With one line from $(0,1)$ to $(1,0)$ and the second line from $(1,0)$ to $(2,1)$. The area under the lines sum up to $1$ and all values of ...