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

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20 views

How to compute the average power of an ergodic process?

Rxx(0)=3 is the average power and if i take limit as t goes to infinity i will get the (E[x])^2 to get variance you subtract 3-2 = 1 is this correct ? and can someone tell the difference ...
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1answer
7 views

Variance function is variance stabilising

Y has mean u and variance function V(u). If $V(u) = \alpha.u^v$ then $h(y) = y^{(2-v)/2}$ is variance stabilising which means that Var(h(Y)) is approximately constant. I tried to prove it computing ...
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1answer
18 views

Finding an interval estimate for $\mu$ given a sample size and variance

I'm in a statistics class and am doing a problem for homework about confidence intervals. I don't really know what it's asking though or when I've even reached a valid solution. The problem says: ...
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2answers
30 views

Expectation over multiple variable?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+k,$$ where $h$ and $k$ are independent random variables with variance ...
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1answer
31 views

How to calculate the probility of 2 independent events of having same value?

We are learning to calculate the probability of sums and difference of random numbers. Here is the problem: One athlete knows from past experience that the distances of his javelin throws follow a ...
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2answers
33 views

Uniform distributed variable X:U(-9,9) is given. Find the CDF of Y if Y is..

$$Y= \begin{cases} 4X,\ \ \ |X| \leq 3 \\ 0,\ \ \ \ \ \ |X|>3 \end{cases}$$ My take on it: $$F_Y(y)=0; y\leq-12;F_Y(y)=1; y\geq 12;$$ $$F_Y(y)=\{Y < y \}$$ In class in a similar task we ...
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1answer
48 views

Proof conditional probability formula

a question for my homework for probability goes as follows: Given X,Y,U, three discrete random variables, prove the following: $$ p_{X|Y}(i|j) = \sum_{k=0}^{\infty}p_{X|YU}(i|j,k)p_{U|Y}(k|j) $$ The ...
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2answers
35 views

Probability distribution of number of columns that has two even numbers in a chart

We distribute numbers $\{1,2,...,10\}$ in random to the following chart: Let $X$ be the number of columns that has two even numbers. What is the distribution of $X$? My attempt: ...
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1answer
59 views

Random Variable $X= U:(0,4)$ is given. Find the CDF of $\min \{ |X-1|, 5-X\}+1$

Random Variable $X= U:(0,4)$ is given. Find the CDF of $Y=\min \{ |X-1|, 5-X\}+1$ X has uniform distribution. So we know that $$Y\in (1,6)$$, therefore $$y\leq 1 \ \ \ F_y(y)=0 ; y\geq6\ \ \ ...
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0answers
7 views

specific examples of random variables satisfying a given condition.

Theorems such as the central limit theorem only says random variables satisfying certain conditions have some properties. Now, what I am curious about is the existence of such random variables. For ...
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1answer
36 views

Expected value of doubling or halving a number with equal probability

I have this question that you start with a value say c. At each step, you either double or half the value with equal probability. Let $X_i$ be the value of c at ith-step, I need to find the expected ...
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3answers
53 views

Finding Variance and Expectation of Boolean Variable

Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is "average of squares of difference from mean ...
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3answers
49 views

What allows us to divide a random variable into multiple ones?

I can't wrap my head around the solution presented for this problem: Suppose a trial has a success probability $p$, let $X$ be the random variable for the number of trials it takes to stop at $r$ ...
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2answers
30 views

transformation of single random variables with absolute value ??

integral I got the final answer to be fy(y)= 1 0< y < 1 I am not sure could anyone correct me if its wrong !
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0answers
23 views

problem about graph of auto-correlation for wide-sense stationary process?

I have the answers but I don't understand the idea and how it can be solved ? please clarify and help me to understand it thank you all
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0answers
24 views

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
34 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
10 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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0answers
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
21 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
42 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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0answers
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
28 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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0answers
11 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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45 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
13 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
18 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
21 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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25 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
34 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
38 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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0answers
37 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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0answers
31 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
75 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
40 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
39 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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0answers
51 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
27 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
31 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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0answers
33 views

Proof of Levy's theorem without Ottaviani inequality [duplicate]

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
31 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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2answers
46 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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0answers
105 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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0answers
16 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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205 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 ...
2
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2answers
44 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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0answers
36 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
34 views

Dice roll, estimator, epsilon

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