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

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Distribution for Arithmetic Mean of n Geometrically Distributed Random Variables

For the evaluation of an algorithm I implemented for work, I need to find the distribution function for the arithmetic mean of $n$ independent, geometrically distributed random variables. Let ...
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1answer
31 views

Does the following sequence of random variables converge?

Let $X_1,X_2,...$ be independent random variables with $P[X_n=0]=1-1/n$, $P[X_n=1]=1/2n$, $P[X_n=-1]=1/2n$ Does $X_n$ converge almost surely? , Does $X_n$ converge in probability? I just started to ...
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1answer
27 views

find the power of a random process?

I know all the steps expect the last step i don't know how to evaluate the integral can someone show me the step that lead to the answer to be A^2/2
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1answer
8 views

What can we say about the concentration around 0 linear transformation of Gaussian random variables?

I have a matrix $X \in \mathbb{R}^{n \times m}$ such that each $A_{ij}$ is a Gaussian with mean $0$ and variance $1$. We have $m > n$. I also have a vector $v \in \mathbb{R}^m$ such that $||v||_2 ...
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20 views

Probability of having at least one coupon out of N types

I'm facing a question regarding random variables: A coupon website has N distinct kinds of coupons. Each selection of a coupon is equally likely and selections are independent. Let $T$ be a ...
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1answer
20 views

Maximum of two independent uniform random variables

Let $x$ and $y$ be uniformly distributed, independent random variables on $[0,1]$. What is the probability that the maximum between $x$ and $y$ is less than $1/2$ and greater than $1/3$?
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1answer
40 views

Probability of most frequent occurrences of suits/values when drawing 4 cards from 52

Draw 4 cards from a card deck with 52 cards (4 colours and 13 values for each colour) one after the other -- none is put back. Let's have two discrete random varaibles X and Y. X counts the maximum ...
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16 views

Find the PDF of $Y = a*X - b*X^3$ given that $X$ is a uniform random variable on $[0,1]$

Assume that $X$ is a uniform RV on $[0,1]$ and that $a$ and $b$ are both positive. Can also assume that $Y$ is monotonically increasing over its range. I'm trying to find the PDF of $Y$ and am ...
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1answer
24 views

Let X, Y be independent random variables and each one has E(1)(exponential distribution). Prove that $W_1, W_2$ are independent.

$W_1=\min \{X,Y\}$;$\ \ W_2=X-Y$ It's given that the density functions for $(X,Y),W_1$ and $W_2$ respectively are: $$f_{(X,Y)}(x,y)=e^{-(x+y)};f_{W_1}(u)=2e^{-2u}, u>0,f_{W_2}(v)={e^{|v|}\over ...
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9 views

What is the expectation of semi-fixed-points in a random permutation?

1<=i<=n is a semi-fixed point if: |π(i)-i| <= 1 with π of {1...n} What is the expectation of semi-fixed point?
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43 views

Expectation of $v=\inf \{n\geq 2\,;\, X_n > X_1 \}$ when $(X_n)$ is i.i.d. uniform on (0,1)

Let $W$ be the occurence meaning the following ordering : $X_1...X_k$ where $X_k$ is greatest.. $X_k$ is greatest, and next in order is $X_1$, and the order of the others is not important. Because of ...
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0answers
11 views

Compute the expectation of a function of a random vector not knowing the whole distribution

Imagine I have three random variables $X,Y$ and $Z$. I know that $X\sim Y$ which does not imply that $(X,Z)\sim(Y,Z)$ I know the distribution of $(Y,Z)$. So, in a summary: I know the distribution of ...
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1answer
15 views

When two random variables that have the same law… Can they be happily exchanges?

Imagine, $X$ and $Y$ are two random variables which have the same law, which we denote by $X\sim Y$. We have then a third random variable $Z$. Can we say that $$(X,Z)\sim (Y,Z)?$$ In what cases is ...
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1answer
19 views

How do I find the following? [Random Variables]

$U-$uniform distribution. $X=U(0,1), Y=U(0,1),Z=\sqrt{1+(X-Y)^2}$ Find $F(z)-$function of distribution. What I've done so far: $\phi(x,y)=\{1,(x,y)\in(0,1) \times (0,1); 0, $otherwise$\}$ ...
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23 views

To prove that induced probability measure indeed defines a probability measure

Given two measurable spaces $(Ω_1, B_1)$ and $(Ω_2, B_2)$, a measurable function T : $Ω_1 → Ω_2$ and P is a probability measure on $(Ω_1, B_1)$. $B_1$ & $B_2$ are respective sigma algebras. The ...
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1answer
32 views

How to find the probability of an estimator?

I'm working on some homework and am having a hard time finding the probability of an estimator. The question says that $Y_i$ ~ Bernoulli($p_2$), where $p_2 = 0.1$. The estimator $p_2 = \bar{Y} = ...
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1answer
25 views

Calculating probability of a normal distribution, not getting correct answer

I'm doing a homework assignment and having some trouble matching the correct answers from my professor. As a reference, I'm calculating these answers using R. The question is as follows: Assuming ...
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36 views

independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper: Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...
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26 views

Are convex functions of a random variable themselves random variables?

I was looking at proofs of Jensen's inequality and noticed that they usually assume that for a convex function $g$ and a random variable $X$, the expression $\mathbb{E}(g(X))$ is well defined, which ...
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1answer
69 views

How to construct Poisson process as a random variable in order to evaluate integrals?

I want to find a probability space $\Omega$ that represents Poisson process as $$\Pi : \Omega \to \{A \in \mathcal{P}{(\mathbb{R^+})}\mid |A| = \aleph_0\}$$ Which is a mapping from $\Omega$ to all ...
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1answer
14 views

Expectation of trigonometric functions involving random variables.

This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form: $\cos(\phi + d_\phi)$, $ ...
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1answer
33 views

Distribution of a matrix product $\mathbf{a}^{H}\mathbf{H}\mathbf{b}$

Could someone help prove the following: I have two independent random vectors $\mathbf{a} \in \mathbb{C}^{M \times 1}$ and $\mathbf{b}\in \mathbb{C}^{N \times 1}$. Both $\mathbf{a}$ and $\mathbf{b}$ ...
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50 views

Conditional expectation random variable composed with a meas. function

I know that the following is true and fairly easily proven. Let $Y$ be a random variable and $\varphi$ a measurable function. Let $A$ be a $\Sigma_Y$ measurable set. If $ X (\omega) = \varphi(Y ...
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1answer
26 views

How to find the estimator using random variables in statistics

I'm doing an assignment for homework in my statistics class. I'm having trouble really understanding what is going on when it comes to estimators, and what the estimator of something is given a random ...
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1answer
12 views

Conditional probability with four random variables

Assume A, B, C, and D are i.i.d random variables and k is a fixed constant. I want to find $\textbf{P}(A < B, C, D | D = k)$. How would I go about getting this, in terms of the cdf of these random ...
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1answer
25 views

Prove that if $X$ and $Y$ are independent discrete variables, then $f(X)$ and $f(Y)$ are independent.

Prove that if $X$ and $Y$ are independent discrete variables, for $f: \mathbb{R} \rightarrow \mathbb{R}$, then $f(X)$ and $f(Y)$ are independent. Here is the exact same question. I define ...
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30 views

Proof of Levy's theorem without Ottaviani inequality

Suppose $X_1,⋯,X_n$ are independent r.v., Let $S_n=X_1+⋯+X_n$, I am looking to show that convergence on $S_n$ in probability implies almost sure convergence by showing that $P(\sup_{m\geq ...
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1answer
29 views

An identity for random variables

My professor gave me the following identity for random variables in her office hours, and I cannot find reference to it anywhere whatsoever. Given a random variable $X_n$ and an estimator $X'_n$, ...
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1answer
14 views

How random are numbers from geometric distribution

If I will choose a value $r$ as a random number from uniform distribution, I can be sure, that this value is totally random - because each value is equiprobable. However, what if I will take a value ...
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2answers
40 views

Uniform Distribution - Show an Expression is Uniform on (a, b)

I'm not quite sure how to deal with this problem. I'm thinking it has to do with uniform random variables in that f(x) = 1/(b-a) if a <= x <= b Otherwise, f(x) = 0. The question would be: If ...
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101 views

Sigma algebra generated by a homeomorphic random variable

Let $\Omega = [0,1]$ be our probability space with sigma algebra of borel sets on $[0,1]$ and Lebesgue measure on $[0,1]$. Let Y be a random variable such that $Y(\omega) = Y(1-\omega)$ for every ...
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11 views

Modifying a generator of random numbers from a trapezoidal distribution to include growth and decay rates

I've written a C# random number generator based on page 11 of this paper: http://pubs.usgs.gov/tm/04/c03/tm4-C3_final_508_files/tm4-C3_apdx1_v030813.pdf It works fine but I would like to modify it, ...
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201 views

Conditional expectation, quadratic function, absolute value

We are given two random variables defined on $[0,1]$: $$X(\omega) = 2 \omega -1 + |2 \omega -1|$$ $$Y(\omega) = 1-|2 \omega^2 -1|$$ I am supposed to find $\mathbb{E}(X|Y)$ which by definition is a ...
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2answers
34 views

is there a concept of asymptotically independent random variables variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...
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29 views

independence copula diagonal

I'm reading Nelsen's Instruduction to copulas, and there is (probably very simple) excersice I cannot deal with. It says that if the diagonal section of the copula equals the diagonal of independence ...
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1answer
29 views

Dice roll, estimator, epsilon

We roll a non-symmetric die. Let $X_n$ be the reulst of $n$-th roll. $$P(X_n = 6)= \frac{1}{6} + \varepsilon, \ P(X_n = 1) = \frac{1}{6} - \varepsilon, \ P(X_n=2) = ... = P(X_n = 5) = \frac{1}{6} $$ ...
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242 views

Show the probability that the sum of these numbers is odd is 1/2

Setting Let $S$ be a set of integers where at least one of the integers is odd. Suppose we pick a random subset $T$ of $S$ by including each element of $S$ independently with probability $1/2$, Show ...
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1answer
10 views

Distribution of sine composed with a random variable

Could you tell me if my calculations are correct? We are given a random variable with the following discrete distribution $$P(X=n) = \frac{2^n}{3^{n+1}}, \ \ n \in \mathbb{N}.$$ Find the ...
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1answer
31 views

Example random variable $\xi$ such that $\xi$ and $\xi^2$ are independent

Find random variable $\xi$ such that $\xi$ and $\xi^2$ are independent.
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25 views

cdfs $F$ and $G$ of random variable $X$, $F\le G$. What can we say about $\mathbb{E}_F[X]$ and $\mathbb{E}_G[X]$?

Problem: A random variable $X$ is distributed in $[0,1]$. Mr. Fox believes that $X$ follows a distribution with cumulative density function $F:[0,1]\to [0,1]$ and Mr. Goat believes that $X$ follows a ...
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1answer
17 views

$X$ and $Y$ are unformly distributed in $[0,1]$ with $P(\max(X,Y)≤z)=P(\min(X,Y)≤(1−z))$. Find $z$.

Problem: Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[0,1]$. For a $z \in [0,1]$, we are told that $P(\max(X,Y)\le z)=P(\min(X,Y)\le (1-z))$. Then, what is ...
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22 views

Inheritance of independence of random variables

I want to show the following statement: let $(X_n)$ and $(Y_n)$ be sequences of random variables and $X_n\perp Y_n$ for each $n$. If $X_n\to X$ and $Y_n\to Y$ in probability respectively, then $X\perp ...
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19 views

If a sequence of random variables all have the same mean, is the sequence tight?

Suppose $(X_n)$ are almost surely non-negative random variables all with the same finite mean $\mu$. Is this sequence necessarily tight?
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44 views

Sigma algebra generated by an absolute value random variable

I need to find out what the sigma algebra generated by $Y$ looks like for $$Y: [0,1] \ni (\omega) \to 1- |2\omega -1| \in \mathbb{R}.$$ The graph of $Y$ is symmetric with respect to $\omega = ...
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1answer
18 views

Meaning of the random variable Y=|X|

I am learning this and having a very basic doubt. Suppose $X$ and $Y$ are two random variables where $X$ takes the values $-2,-1,0,1,2$ each with probability $1/5$ and $Y=|X|$. I think $Y=|X|$ means ...
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22 views

Distribution of Logistic of Normal

If $X \sim N(\mu_X, \sigma^2_X)$ and $Y= \frac{\exp(X)}{1+\exp(X)} $, what is the distribution of $Y$? I thought logit-normal would fit the bill, however the logit of $Y$ is ...
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9 views

Do derivative and expectation operators commute only for discrete random variablesÉ

Suppose we have a function $$g(r)=E(f(r,X))$$ where $X$ is a random variable and $g:\mathbb{R}\rightarrow\mathbb{R}$. If $X$ is a discrete random variable, we can simply write $$g(r)=\sum_{x\in ...
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27 views

conditional expectation given two conditions

I want to check my understanding of conditional expectation. Could someone confirm if this is true? Y(t) is normally distributed. E[Y(2)|Y(1),Y(3)] = E[Y(2)|Y(1)] + E[Y(2)|Y(3)] If this is not ...
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1answer
38 views

An equivalent condition for a random variable to be integrable

I have to prove the following fact. Show that $X_1$ is integrable, iff for all $\epsilon>0$ $$\sum_{n=1}^{\infty} \mathbb{P}(|X_1|>n \epsilon)<\infty.$$ Here $X_1$ is just a random ...
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1answer
14 views

Issue with sum of probabilities of probability distribution function of a geometric random variable

Is it possible that the sum of probabilities of geometric distribution for "$k = 1,...,n$", where k is number of trials until the first success, is not equal to 1? I'm asking this, because I encounter ...