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

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3answers
23 views

if Y1 and Y2 are independent, does it follow that U and X are independent, where U = f1(Y1, Y2) and X= f2(Y1, Y2)

If X, Y are iid rvs, and U and Z are r.v.s that can each be written in terms of X and Y, does that mean that U and Z are independent?
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0answers
35 views

How to calculate the mutual information between two outputs of Rayleigh fading channels

We have the two channels: $$X_{a,i} = H_{i}s_{i} +N_{a,i} \\ X_{b,i} = H_{i}s_{i} +N_{b,i} $$ for $1 \leq i \leq n$, where $H_i$ denotes the i.i.d. channel coefficient and is a zero-mean complex ...
2
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0answers
29 views

PDF of Random Variable $\sin\alpha \cdot \cos\beta$ with $\alpha,\beta$ uniform

As part of a bigger problem, I want to compute the probability density $f_Z(z)$ of $$Z = \sin\alpha \cdot \cos\beta$$ where $\alpha, \beta$ are random variables, independently and uniformly ...
2
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1answer
63 views

Convergence of descending sequence of random variables

I have this for homework: Let $X_1, X_2,\cdots,X_n, \cdots$ be descending sequence of random variables. Show that $X_n$ converges to 0 in probability if and only if $X_n$ converges to 0 almost ...
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0answers
35 views

A question about weak convergence of random variables

I am reading my lecture notes and our definition of weak convergence or random variables is: First another definition: A sequence $\mu_n$ of probability measures on $\mathbb R$ converges weakly to a ...
1
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1answer
31 views

Convergence of random variables: struggle with a proof

I am trying to understand a proof of the following theorem: $X_n$ is a sequence of random variables. $X_n \to X$ in probability $\implies$ that each sub-sequence of $X_n$ has a sub-sequence which ...
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1answer
51 views

Joint Probability and Intersection Probability

Given two independent events A and B: $P(A \cap B)= P(A)*P(B)$ but then I saw somewhere that: $P(A \cap B)= P(A)*P(B)= P(A|B)*P(B) = P(B|A)*P(A)$ where for example $A$ is $X=x$ and $B$ is $Y=y$ ...
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1answer
36 views

How to find $\mathbb P(XY<\frac{1}{2})$ and $\mathbb P(Y< X^2)$ without convolution?

Let $X$ and $Y$ be uniformly distributed on $[0,1]$ and independent random variables. Find $$\mathbb P\left(XY<\frac{1}{2}\right) \text{ and } \mathbb P\left(Y< X^2 \right).$$ Tip: one can do ...
1
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1answer
33 views

Expected value with probabilities

Joel owns a lawn care business and recently performed some research on the size of 50 lawns that he takes care of. Joel recalls that he is expected to take care of a total of $21$ acres of lawn for ...
1
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2answers
37 views

Verify the joint probability function

I had a question I was hoping for some help on: There are 8 similar chips in a bowl: 3 marked (0;0), 2 marked (1;0), 2 marked (0;1), and 1 marked (1;1). A player selects a chip at random and is given ...
3
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1answer
36 views

Expected value of an expected value of a joint density function

I had a question I was hoping for some help on: Let $Y_1$ and $Y_2$ be continuous random variables with joint density function: $$f(x,y) = \begin{cases} 6(1-y_2) & \text{if $0 <= y_1 <= ...
2
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0answers
13 views

Is AR(p) strictly stationar?

Good evening. Is it true that model AR(p) is a strictly stacionar random sequence? Model AR(p) is given by $X_{t} = \varphi_{1} X_{t-1} + \ldots + \varphi_{p} X_{t-p} + Y_{t}$ where $\{Y_{t}\}_{t \in ...
3
votes
1answer
38 views

Writing random variable formulas with set notations, What is the problem?

Is it wrong to write $\displaystyle P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}$ when $X$ and $Y$ are random variables? As I know a random variable is a function and therefore has a range and the two ...
3
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1answer
45 views

Prove $E\left[\left(\frac{1}{n}\sum_{i=1}^nX_i\right)^k\right]\leq E\left[X_1\left(\frac{X_1+(n-1)\mu}{n}\right)^{k-1}\right]$

$X_1,X_2,\ldots,X_n$ are i.i.d. random variables, $X_1>0$, $E[X_1]=\mu$, $E[X_1^k]<\infty$ for $1<k \leq2$. Prove: $$ E\left[\left(\frac{1}{n}\sum_{i=1}^nX_i\right)^k\right]\leq ...
2
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1answer
70 views

Question about asymmetry of chi-square distribution

Let $X_1,\dots,X_n$ be a set of i.i.d. chi-square random variables with $k$ degrees of freedom. Consider the statistic $\arg\max_i\{|X_i/k - 1|\}=X_{\alpha}$. I wonder about the probability that ...
2
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1answer
33 views

Problem solving the standard deviation for a stochastic variable

Information: In a laboratory we have a vial of water that's being kept on 50 degrees Celsius. This is measured with the best thermometer in the world. The standard deviation on this thermometer is ...
2
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2answers
41 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
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1answer
28 views

Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
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1answer
15 views

CDF of the kinetic energy of a particle under uniform distribution

We are given that X~Uniform[2,3] and the kinetic energy is $T=\frac{1}{2X^2}$ I tried the following: $P(T\leq a) = P(\frac{1}{2X^2}\leq a) = P(-\sqrt{2a}\leq X \leq \sqrt{2a}) = ...
2
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1answer
41 views

Find the covariance of $Y_1$ and $Y_2$

I had a statistics question I was hoping for help on: Let $Y_1$ and $Y_2$ be discrete random variables with join probability function: $$f(x,y) = \begin{cases} \dfrac{y_1 + 2y_2}{18} & \text{if ...
1
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1answer
39 views

Independent events and Kolmogorov

Suppose we have a probability space $(\Omega, \mathfrak{F}, P)$, and independent events $(E_n)_n$. Consider $$M_n = \sum_{k=1}^n I_{E_k}$$ Is it correct to say that by the Kolmogorov $01$ law $M_n$ ...
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1answer
51 views

Zero variance Random variables with density [duplicate]

I found here the question: Can a random variable have a density function whose variance is $0$ ? I understood as a random variable which has a density. What is your opinion on what I ...
2
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2answers
42 views

Convergence a.e. of the series $\sum_{i=1}^{n^2} \frac{X_i}{n^2}$

Let $(X_n)_{n\geq 1}$ be independent random variables with expected value $m$ and $\sup_n Var(X_n)\leq K < \infty$, and they are uncorrelated. Then $1)$ $$\sum_{i=1}^n \frac{X_i}{n} $$ ...
2
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2answers
141 views

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n ...
0
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1answer
54 views

Expected valued of Random sums about dice and jar problem

A six-sided die is rolled , and the number N on the uppermost face is recorded. From a Jar containing 10 tag numbered 1,2,,,,10 , we then select N tags at random without replacement. Let X be the ...
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0answers
9 views

Random Variable 1

This is my answer: E[X]=1(0.4)+2(0.2)+3(0.1)+4(0.3) =0.4+0.4+0.3+1.2 =2.3 E[W(X)]=E[200-10X] =E[200]-E[10X] =200-10(2.3) =200-23 =177 and i don no how to find ...
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0answers
51 views

Expectation of matrix product

Suppose we have a random matrix $M \in \mathbb{R}^{n\times m}$ such that $\text{E}[M] = 0$ and $\text{E}[M M^\top] = \Sigma$. How does one compute $\text{E}[M^\top M]$?
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1answer
44 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
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0answers
44 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
2
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1answer
93 views

Probability of Random Event and Conditionality

A company has been running a television advertisement for one of its new products. A survey was conducted. Based on its results, it was concluded that an individual buys the product with probability ...
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1answer
67 views

sum of two dependent random variables

Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$. $X$ and $Y$ ARE NOT ...
2
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1answer
55 views

Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$

This is an exercise from Williams, Proability with martingales. Prove that if $Z$ has the $U[-1,1]$ distribution, then $$\phi_Z(t) = \frac{\sin t}{t}$$ Then prove that do not exist IID random ...
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0answers
13 views

transformation and functions of random variables

Let $X,Y$ be independent random variables. I already have the distribution of $XY$ over a certain subinterval of $\mathbb{R}$, by convolution. My question is, is it possible to get the distribution of ...
2
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1answer
49 views

$\frac{S_n}{n} \to -1 \ \ a.e.$, exercise from probability book

I'm stuck with this exercise from Williams, probability with martingales. Let $X_1, X_2, \ldots $be independent random variables with $$P(X_n = n^2-1 )= \frac{1}{n^2}$$ $$P(X_n = -1 )= ...
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1answer
103 views

Exercise from Williams book Probability with martingales

I'm doing this exercise from Williams book Probability with martingales Let $(X_n)$ be a sequence of IID random variables with $E(|X_n|) = \infty $ for all $n$. Then prove that $$1)\ \sum_n ...
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0answers
57 views

Probability distributions with closed-form cumulative distribution functions (CDFs)

I am interested in finding multivariate probability distributions for which the cumulative distribution functions (CDFs) are given in close form. For instance, the multivariate Gaussian distribution ...
2
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1answer
46 views

Summation of binomial number of poisson random variables

Z is summation of K random variables that each has Poisson distribution with different means. But, K is a Binomial random with parameters of n and p. I was wondering what is the distribution of Z?
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1answer
31 views

Distribution of random variables when combined

I need help with this problem: If $X$ and $Y$ are two independent random variables and are both standard normal, what is the distribution of $\frac{1}{2}(X^2+Y^2)$? I think I start with ...
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1answer
78 views

Probability that sum of two uniformly distributed random variables is less than some constant

I am trying to find a way of determining the probability that the sum of two continuously uniformly distributed random variables is less than some constant $C$, formally: Let $A \sim ...
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2answers
46 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
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0answers
35 views

sum/product combination of random variables

Let $X$ and $Y$ be independent random variables. If I am asked about the distribution of random variable $XY+Y$, is it ok if I compute $XY$ first and then add the result to $Y$ (via convolution, or ...
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0answers
21 views

What is the nonlinear estimator for Gaussian Random variable?

I know that the best estimator is $g(x)=E\{Y|X=x\}$ and the conditional density for jointly Gaussian random variables is known to be Gaussian with mean and variance given by ...
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0answers
18 views

A stochastic process is generated as follows: we assign the value 1 to a head and the value 0 to a tail. Start at n=0, Compute Rxx(0,0) and Rxx(2,3)

I am kind of confused here, since autocorrelation describes the correlation between values of the process at different times, but for the first case, it is at the same time. I got that ...
3
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1answer
54 views

$X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$

Suppose $X, Y$ are random variables in $L^2$ such that $$X = E(Y | \sigma(X)) $$ $$Y = E(X | \sigma(Y))$$ Then I want to show that $X=Y$ almost everywhere. What I've done: By conditional Jensen ...
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0answers
41 views

Probability question involving drawing balls from an urn

Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...
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1answer
34 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
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0answers
27 views

Mean Preserving PDF Spreading

I have a histogram representing the PDF of an unknown discrete RV. The histogram is asymmetrical. To be clear, all I have is the histogram. Is there a known way to increase/decrease the variance of ...
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3answers
64 views

Product of random independent variables

What are the properties of the product of random variables? The book I have on probability and statistics only comments on their sum properties. I know that when you sum random independent variables ...
2
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2answers
179 views

Expectation value of absolute value of difference of two random variables

I do not really know how to prove the following statement: If E(|X-Y|)=0 then P(X=Y)=1. The main problem is how to handle the absolute value |X-Y|. If I say that |X-Y| >= 0 it follows that ...
0
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2answers
109 views

How to compute the sum of geometric distribution [closed]

How to compute the sum of random variables of geometric distribution $X_{i}(i=0,1,2..n)$ is the independent random variables of geometric distribution, that is, $P(X_{i}=x)=p(1-p)^{x}$, then how to ...