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

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1answer
14 views

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are ...
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0answers
16 views

Is this a misuse of the term “probability space”?

Let me first state the definitions as I am using them. Do correct me if I am wrong here! A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The ...
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1answer
24 views

Distribution of Summation of two discrete random variables

Here, $X_1$ and $X_2$ are independent discrete random variable and the support set of $\tilde{x}_1$ and $\tilde{x}_2$ respectively. We have mentioned the support sets below: $$ X_{1} = \{ 2,3,...,7 ...
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0answers
21 views

Random process question. How to find the mean,autocorrelation and WSS? [on hold]

A fellow student posted this question on our whatsapp group for the course. A random process $$X(t) = A\cos(\omega t)B\sin(\omega t) $$ where $A$ and $B$ are random variables. Find the ...
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2answers
24 views

The distribution of the product of Gaussian variable and Rademacher variable.

I have two independent variables: $X$ follows from standard Gaussian distribution $N(0,\sigma^2)$; $Y$ follows from Rademacher distribution, i.e., $Y$ can be either $-1$ or $1$ with the same ...
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1answer
33 views

Determining Probability Generating Function from Probability Mass Function and Convergence

I am trying to solve the following: Suppose $X_{nk}, k=1,2,\ldots,n, n≥ 2$ are i.i.d. random variables $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
1
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1answer
53 views

If X,Y and Z are independent, are X and YZ independent?

If yes: I know that f(X) and g(Y) are independent if X and Y are independent and f and g are "measurable".* If that is to be used, is g(Y) = YZ measurable? If not, how else to approach this? If ...
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1answer
31 views

Application Problem: Expectation and Variance of Compound Poisson Process

I am solving the following: Let $Y1, Y2,…$ be a random sample from $\Gamma(p,a)$ distribution, where p and a are positive real numbers. $Y$ is damage in thousands of dollars caused to a car in an ...
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0answers
12 views

Is the product of two sub-Gaussian random variables a sub-Gaussian random variable?

If not, is there any way to make it hold? Note: the random variable $x$ is called $σ^2$-sub-Gaussian if $E[e^{tx}]≤e^{t^2σ^2/2}$.
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1answer
25 views

Power Spectral Density Approximation

Let $X_t$ be a zero-mean, stationary random process. Let $X_f$ be the Fourier transform of $X_t$; $X_f$ is also a random process, but as a function of $f$. Let us denote the power spectral density ...
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0answers
35 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{\frac{5}{9}}$ in distribution, where $Z_n ...
4
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1answer
132 views
+50

Impact of random numbers on the eigen-values

How do the eigen-values of the following tridiagonal matrix ($A$) change when adding random numbers $R_i$ (with a normal distribution with the mean 0 and variance $m$) to its diagonal. A is a square ...
1
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2answers
19 views

Comparison between two exponentail random variables

A and B are exponentially distributed with parameter $\alpha$ and $\beta$. A and B race repeatedly. $N_b$ denotes the number of times B wins before A wins his first race. Find $P (N_b = n )$ for $n ...
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2answers
46 views

Showing that infinite product of random variables goes to zero: $\prod^\infty X_i \rightarrow 0 \text{ a.s.}$

I am doing the following exercise: Let $X$ be a strictly positive rv with $\mathbb E[X]=1$ but $X \neq 1$ almost surely. Let $X_1, X_2 \dots$ be iid with same distribution as $X$. Now let $M_0=1$ and ...
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1answer
25 views

CDF and Convergence of Maximum of Sequence of i.i.d. R.V. of Random Length

Let $X_1,X_2,...$ be i.i.d random variable $U(0,1)$ distributed. Let $N_m$ be $Poisson(m)$ and independent of each $X_i$. i)Find the cumulative density function of ...
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1answer
29 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
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1answer
24 views

Working with the random variable $\log X$ instead of $X$

Suppose I have a positive stochastic process $X_t$. I'd like to compute certain properties about $X_t$, but suppose I can't and instead I can compute properties about $\log(X_t)$. Can I say anything ...
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0answers
15 views

How to deal with set-valued(set in $\Re^n$) random variables?

I'm trying to attack a problem where the random variable are sets i.e set-valued random variable. Suppose $S = \{X_1, X_2,\cdots,X_n\}$ is a set of sets($X_i$) and $f(X_i)$ is the probability ...
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1answer
35 views

Deriving Probability Density Function from Probability Generating Function for Random Sum

I am trying to solve the following: Let $X_{i}$~$Geometric(q) i=1,2,...,N$ with $q=1-p, 0<p<1$. $N$~$Geometric(p)$. Define $Y=\sum_{i=1}^{N}X_i$ and assume each $X_i$ is i.i.d. and ...
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1answer
29 views

$E(f(|X_n|))$ property implies uniform integrability? [on hold]

This is exercise 6.10 in Resnick's book "A Probability Path". We're given a sequence of random variables $(X_n)$ and an increasing function $f: [0, \infty) \rightarrow [0, \infty)$ such that $$ ...
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1answer
41 views

Showing Convergence in Distribution for Conditional Random Variable

I am trying to prove the following: Let $X$ and $Y$ be random variables such that $Y | X = x$ ~ $N(0, x)$ with $X$ ~ $Po(\lambda$). Show that $\frac{Y}{\sqrt{\lambda}} \to N(0,1)$ in distribution as ...
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1answer
40 views

Computing Conditional Characteristic Function

I am trying to compute the characteristic function of the following: Let $X$ and $Y$ be random variables such that $Y\mid X = x\sim N(0, x)$ with $X\sim\mathrm{Po}(\lambda)$. Find the characteristic ...
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1answer
40 views

How can I find the density of $E[X\mid Y]$ when $(X,Y)$ is gaussian

I was tying to prove the following: Given $(X,Y)$ a centered gaussian vector in $\mathbb{R}^2$ with the following covariance matrix $$ \Sigma = \begin{bmatrix} \sigma^2_x & \sigma_{x,y} \\ ...
2
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1answer
69 views

Showing Convergence in Distribution of Continuous Function of Sums of R.V.s

I am trying to solve the following: Let $X_1, X_2, . . .$ be i.i.d. r.v.s with mean $\mu$ and positive, finite variance $\sigma^2$, and set $Sn = \sum_{k=1}^{n} X_k, n ≥ 1$. Suppose that $g$ is twice ...
3
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1answer
46 views

Proving a Variation of the the Central Limit Theorem

I am trying to prove the following: Let $X1, X2, . . .$ be positive, i.i.d. r.v.s with mean $\mu$ and finite variance $\sigma^2$, and let $S_n = \sum_{k=1}^{n} X_k$ , $n \ge 1$. Show that $\frac{S_n ...
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0answers
31 views

Show that $X_1,X_2, \ldots$ are identically distributed [closed]

Suppose $A_1,A_2,\ldots$ are i.i.d. with positive expected value and $$S_0=0, S_n=S_{n-1}+\min\{t>0: A_{S_{n-1}+1}+\cdots+A_{S_{n-1}+t}<0\}.$$ Let $\tau=\min\{n:S_n=\infty\}$. Let ...
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0answers
17 views

Random variable with respect to the event space [closed]

Having the probability space $(S, \mathcal F, \Pr(\cdot))$ and a very large $\mathcal F$, like the power set, how do we define a function $X(\cdot): S \rightarrow \mathbb{R}$ which is not a random ...
2
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1answer
35 views

How do you proof that F is a distribution function, when x > 0

I hope that someone could help me solve this question of my textbook: Let F (x) = e^(−1/x) for x > 0 and F (x) = 0 for x ≤ 0. Is F a distribution function? If so, find its density function. How do ...
1
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1answer
36 views

Convergence in Probability for a Sequence of Random Variables

I am trying to solve the following: Let $\{X_n, n ≥ 1\}$ be a sequence of i.i.d. random variables with density $f(x) = e^{−(x−a)}$, for $x ≥ a$ and $f(x) =0$, for $x < a$. Set $Y_n = \min(X_1, ...
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0answers
24 views

FFT Hyperbolic Distribution R

This is my first posting so forgive me if it is not 100% in line with this forum's best practices. I am completing an analysis using ICA as the decomposition technique. I am keeping 4 of the 10 ...
2
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1answer
25 views

Computing Distribution of Conditional Expectation of Gaussian RV

I am trying to compute distribution of the following random variable \begin{align*} E[(X-E[X|Y])^2|Y] \end{align*} where $X \sim \mathcal{N}(0,\sigma^2_x)$ and $Z \sim \mathcal{N}(0,\sigma^2_Z)$ where ...
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0answers
41 views

What is the product of two independent random variables (as mentioned below)?

Let $X$ and $Y$ be two random variables with: $\begin{equation} f_{X}(x) = \begin{cases} e^{-\lambda T} & \text{if } x = 0;\\ \lambda T e^{-\lambda T(1-x)} & \text{if } 0 < x \leq ...
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0answers
15 views

Is this expression true for moments of random variables?

Suppose $X_1(t), \cdots, X_n(t)$ are random variables of a continuous time stochastic process. Suppose for any $p>1$, $\sup_{t \geq 0} E\left[\sum_{i=1}^n X_i(t)^p \right] < K_p$ where $K_p$ is ...
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0answers
33 views

The definition about 'with high probability (w.h.p.)' [closed]

w.h.p. can often be seen in the analysis of randomized algorithms. It's definition can be seen here https://en.wikipedia.org/wiki/With_high_probability. However my confusion is that: Assuming we ...
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0answers
34 views

determining if it is a random variable

I know that $\int_0^{\infty}e^{-\alpha t}c(X_t)dt$ is a random variable when $c(.)$ is a measurable function and $X_t$ is a stochastic process. How can this be proved rigorously?
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0answers
24 views

closed form p.d.f of euclidean norm of random variables

If $X\sim N(0,\sigma_1^2)$,$Y\sim N(0,\sigma_2^2)$,$Z\sim N(0,\sigma_3^2)$ and given that X,Y,Z are independent random variable with normal distributions, then the random variable $U=\sqrt{X^2+Y^2}$ ...
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0answers
8 views

How to define a one-parameter family of probability distributions

I am trying to evaluate a noise-source as a means of providing entropy to a random number generator. I am running into trouble when it comes to determining the probability distribution that has the ...
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1answer
18 views

Expectation of normal and log normal distribution

Let $X \sim N(\mu_x, \sigma_x^2)$ and $Y\sim N(\mu_y, \sigma_y^2)$, with correlation $\rho$. How do I find $$E[Xe^Y]$$? I tried a bunch of things without result. I'm also interested in "general" ...
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1answer
33 views

Application Problem: Random Sums of Random Variables and Correlation

I am trying to answer the following: The number of traffic accidents per year at a given intersection follows a Poisson(10, 000)-distribution. The number of deaths per accident follows a ...
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2answers
36 views

Var$(X) = \mathbb{E}((X - \mathbb{E}(X))^2) = \mathbb{E}(X^2) - \mathbb{E}(X)^2$

I have a question about something my teacher told us: let $\mathbb{E}$(X) donate the expected value of a certain random variable $X$. Then Var$(X) = \mathbb{E}[(X - \mathbb{E}(X))^2] = ...
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4answers
44 views

Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$.

I'm having trouble with solving this problem: Roll a die until, for the first time, the same side shows up two times in a row. Let $X$ be the number of rolls needed. compute $\mathbb{E}(X)$. I know ...
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2answers
31 views

Take $k$ shoes ($k \leqslant n$) from a wardrobe. What is the expected value of the number of pairs ($X$) you take?

I'm having trouble with this question: Let there be $2n$ shoes ($n$ pairs) in a wardrobe, arbitrary ordened. Take $k$ shoes ($k \leqslant n$) from that wardrobe. What is the expected value of the ...
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0answers
28 views

“Distance” between two distributions

Hello, I need help understanding this problem. I have no idea how I'd approach it and have some problems following the solution. I can understand the first line, how $F_1$(x)=x because it increases ...
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1answer
42 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I ...
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2answers
42 views

time it takes to service a car with exponential random variable with rate 1

Need help with this question here. Ill post exactly what it says then show my ideas so far. "The time it takes to service a car is an exponential random variable with rate 1. (a) If A brings his car ...
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1answer
23 views

Given the probability density of random variable $X$, what is the density of $Y=aX+b$?

I have a random variable $X$ with probability density $f_X$ and want to determine the probability density $f_Y$ of $Y=aX+b$ with $a,b \in \Bbb R$. How do I proceed here?
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1answer
41 views

Finding what distribution a random variable has.

Mike has a gold coin with fair probability, and a silver coin with $1\over 3$ probability for Heads and $2\over 3$ for tails. He tosses the gold coin 120 times and the number of heads is denoted N. ...
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2answers
34 views

Comparing two exponential random variables

Let $A$ and $B$ be independent random variables drawn from the exponential distribution with parameters $\lambda_A<\lambda_B$. What is the probability that $A<B$? I'm of course aware of the ...
1
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1answer
25 views

Confusion With “Nested” Random Variables

Problem The probability that a compnay's workforce has no accidents in a given month is $0.7$. The numbers of accidents from month to month are independent. What is the probability that the third ...
2
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1answer
21 views

Expectation of cumulative distribution function of a standard normal distributed random variable

Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. The find the expectation of $\Phi(X)$. I ...