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

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1answer
45 views

Continuous mapping theorem - counterexample

The continuous mapping theorem states that Let $g: R^n \rightarrow R^k $ be continuous in every point of a set $C$ such that $\mathbb P\left(X\in C\right)=1$. If $X_n \xrightarrow{d} X $ then ...
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1answer
37 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 $$P(X_{nk}=0)=1-\frac{1}{n}-\frac{1}{n^2}\\P(X_{nk}=1)=\frac{1}{n}\\P(X_{nk}=2)=\frac{1}{n^2}$$ ...
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2answers
224 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
35 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
19 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
32 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
49 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 $Z_n \to N(0,\sqrt{5/9})$ in distribution, where $Z_n ...
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2answers
179 views

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 ...
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2answers
30 views

Comparison between two exponentail random variables

A and B are exponentially distributed with parameter $\alpha$ and $\beta$. A and B race with each other continuously. $N_b$ denotes the number of times B wins before A wins single time. Find $P (N_b ...
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2answers
50 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
28 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
18 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
37 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
40 views

$E(f(|X_n|))$ property implies uniform integrability?

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
44 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
42 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} \\ ...
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1answer
72 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 ...
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1answer
56 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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1answer
38 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 ...
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1answer
37 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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1answer
28 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
46 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
16 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
35 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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10 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
20 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
37 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
45 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
34 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
33 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
46 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
43 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 ...
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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 ...
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1answer
26 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 ...
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1answer
34 views

Density probability function and distribution of $Y=cosX$

I have the following problem. Let X be a random variable uniformly distributed on $[-1,1]$ and let be $Y=cos(X)$. a) Find the function of density and distribution of Y b) Find the espectation of Y ...
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0answers
9 views

Let $X \sim N(0,\sigma^2)$ and $Y$ a positive r.v. Possible to determine sign of $Cov(X,Y)$?

The problem is that $X$ and $Y$ are dependent r.v.s and are dependent via some complicated stochastic differential equation. I'm wondering if it's possible to determine if $Cov(X,Y) \geq 0 $ or ...
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1answer
29 views

Expectation of quotient of linear combinations of independent standard normal random variables

Let $a, b, c, d, e, f$ be complex numbers with nonnegative real parts and nonnegative imaginary parts, and let $X_{1}, X_{2}, X_{3}, X_{4}$ be independent standard normal random variables. How can I ...
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1answer
19 views

Are RV having same exp. value and covariance already have the same distribution?

Let $(X_1, ..., X_n), (Y_1, ... , Y_n)$ be random variables. $X_i$ has the same distribution as $Y_i$ for all $i$. $\forall i, j: Cov(X_i, X_j) = Cov(Y_i, Y_j)$ Do $(X_1, ..., X_n)$ and $(Y_1, .., ...
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1answer
32 views

Cross-correlation of identical sets: not getting expected result

I'm trying to work out the correlation coefficient of two sets using a given formula, but I'm not getting a perfect correlation when using identical sets. The correlation between a client’s ...
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2answers
64 views

The probability density of $X^2$?

Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? ...
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2answers
31 views

Probability: Combinatorics and discrete random variables

I've managed part (i) fine but have no idea how to approach part (ii). I tried using Baye's theorem in order to calculate the conditional probability that the red team has size k given that it ...
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2answers
62 views

How big is the chance that a arbitrary man is taller than a arbitrary woman?

I'm a first year mathematics student, and I'm having trouble with computing the following: Assume that in a country the height $X$ of men is normally distributed, with $\mu_X = 180$ (the expected ...
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0answers
23 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...