-2
votes
0answers
13 views

Proof PDF is a valid function [on hold]

Consider a random variable $Y$ which has probability density function (pdf) defined by $f(y)=(kθ^k)/y^(k+1) ,y≥θ$ and $f(y) = O$ for $y <θ$ , where $θ > O$ and $k > 2$. Show that $f(y)$ is a ...
0
votes
1answer
14 views

Distribution of a function of a uniform random variable.

I ran across this example the other day and was surprised at how stumped I was. Suppose $U$ is a uniform random variable on the interval $[0,1]$. Let $F = \frac{1}{U+3}$. What is: ...
0
votes
1answer
26 views

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...
1
vote
1answer
25 views

Finding density function of random variable, which is division of two other random variables.

I have following 2-dimensional random variable $(x,y)$: $$ f(x,y) = 1, \quad 0 \leq x \leq 1, \quad 0 < y \leq 1 $$ I have to find density function of random variable $Z = \frac{X}{Y}$. I am ...
1
vote
1answer
11 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
-2
votes
0answers
22 views

Help with random variable to found probabilty (PDF)

Stuck in this example to found (PDF) in many conditions
2
votes
1answer
33 views

Probability exercise Bernoulli. [closed]

Probability random signals. Im late I have no idea to start and this is for tomorrow. I was on training and have no break to do this work. I do this.You are an Internet savvy and enjoy watching video ...
1
vote
1answer
120 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
1
vote
3answers
30 views

Probability excersice

If $Z$ is a Gaussian random variable with mean $\mu_Z = 0$ and variance $\sigma^2_Z = 1$, and $Y$ is defined as: $$Y=a + bZ +cZ^2$$ for some constants $a, b, c$ show that the correlation ...
0
votes
1answer
28 views

Equivalence, identity of random variables

Suppose I have $X \sim \text{Uniform}(0,1)$ and $Y \sim \text{Uniform}(0,1)$ As we all know $X+Y$ is a triangular distribution. What of $X+X$? Surely this is uniformly distributed on the interval ...
3
votes
1answer
52 views

Function of a uniformly distributed continuous random variable

Basically, I'd like to add $n$ random vectors in a 2 dimensional space of unit length and of angle $\theta$ relative to a global axis. The probability density function of the angle $\theta$ is a ...
0
votes
1answer
40 views

Probability distribution from mean time to failure [closed]

A factory has different categories of machines which require frequent adjustments and repair. Each category of machine fails uniformly after continuous operation and the failure profile of the ...
0
votes
1answer
23 views

Pdf calculation of two random variables

If $X = aY + b$, both $X$ and $Y$ are random variable. The pdf of $Y$ is given, can anybody please tell how to find pdf of $X$ ?
1
vote
1answer
22 views

For random variables $X$ and $Y$, $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ if and only if $f_{X,Y}(x,y)=f_X(x)f_Y(y)$

I have seen either statement used as a definition of independent random variables. I was trying to prove their equivalence for discrete random variables. I am able to prove that if the joint density ...
1
vote
2answers
54 views

Must every probability distribution over a countable set be discrete?

Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as: Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, ...
0
votes
1answer
29 views

What is the distribution of the dot product of a Dirichlet vector with a fixed vector?

I am trying to get the distribution of a weighted sum when the weights are uncertain: $S = \sum\limits_{i=1}^N w_iC_i = \mathbf{w}\cdot \mathbf{C}$ where vector $\mathbf{w}$ is random with components ...
1
vote
2answers
42 views

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?
0
votes
1answer
25 views

Calculate $E[XY]$ of dependent variables

I'm having a little trouble whit a probabilistic exercise. The problem says this: There's a vase whit 10 marbles, 4 black and 6 white. Now I extract 2 of them without reposition. Being $X,Y$ random ...
0
votes
1answer
17 views

Variance of a linear combinations

I was given the problem above. I am confused on how to find the variance of the linear combinations. A for example would have a mean of 22 correct? Can someone ...
1
vote
3answers
49 views

Random variable with infinite expectation but finite conditional expectation

I've been very stuck on a question from Probability and Random Processes by Grimmett and Stirzaker for ages - so stuck that I flicked to the back to have a look at the answers. But, I can't seem to ...
0
votes
0answers
26 views

MATLAB Poisson r.v. Expected Value Plot

Here is my MATLAB code for P(X>x) for a Poisson r.v. with E(X)=10. I think I am not computing P(X>x) correctly. Can anyone help? clear; clc; U=rand(1,100000); for i=1:numel(U) k = 0; lambda ...
0
votes
1answer
33 views

Given the distribution of a random variable $R$, who do you get a uniform random variable $U$?

Let us say you have a random variable $R$. How would one generate a uniform random variable $U$, with the maximum possible entropy (or infinite entropy, if $R$ has such)? (For simplicity, you may ...
0
votes
1answer
29 views

Moments of Geometric Random Variable

Let $X$ be a geometric random variable i.e. it represents the number of consecutive failures before you get the first success where the success probability is $\rho$. We know $E[X] = 1/\rho$ and ...
0
votes
0answers
15 views

Maximum-Likelihood parameters for Bernoulli trials with decaying success probability

Say I have a sequence of independent bernoulli trials $X_0, X_1, \ldots, X_n$, with exponentially decaying success probability $\mathbb{P}(X_t=1)=r^{-t}$, for some unknown parameter $r$. How can I ...
4
votes
0answers
122 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $Bin(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define recursively ...
1
vote
1answer
84 views

Solution of equation of binomial random variables

Is it possible to find the probability distribution of the random variable $X$ that solves the following equation? $$ X = Bin(X, p) + Bin(X, 1-p), $$ where $Bin(X,p)$ is a random variable distributed ...
0
votes
1answer
31 views

Bounded function of geometric random variable

if X~ Geometric(p), with q=1-p, then show that for any bounded function f with f(0)=0, we have E(f(x)-qf(x)+1)]=0. Our professor asked us to try solving this problem as a good practice but I have no ...
1
vote
0answers
45 views

Accuracy of a Normal Approximation for a Poisson random variable.

compute bound on accuracy of a normal approximation for a poisson random variable with mean 100? I understand what the question is trying to ask me but I have no idea how to approach it and solve it. ...
1
vote
0answers
12 views

How to set covariance for Bivariate Logistic Distribution

This is the logistic distribution of single random variable (taken from Wikipedia). x = random variable mu = mean of all random variables s = variance. Now, I want to do a Bivariate logistic ...
1
vote
1answer
46 views

Random variables with joint density function

Let R be the rectangle $\ \{(x, y); 0 <= x <= 2, 0 <= y<= 1\} $, and let $\ f(x, y) = >k(x^2+ y^2)$ on R and zero elsewhere. (a) Find the value of k which makes f a joint ...
0
votes
0answers
16 views

How can I visualize the sample space of a complicated random experiment?

A sample space $\Omega$ of a random experiment is the set of all possible outcomes $\omega$. When dealing with random variables (which, formally, is a map from $\Omega \mapsto R$), we usually operate ...
0
votes
2answers
45 views

Probability distribution in wireless channel?

Let suppose that I have a random variable $X_{mn}=\sqrt{\left(1/d_{mn}\right)^\alpha}\times h_{mn}$ wherr $d_{mn}$ is a random variable with uniform distribution and $h_{mn}$ is a random variable with ...
1
vote
1answer
31 views

Geometric random variable $X$, $Pr(X\ is\ even) =$?

Original Question: Toss an unfair coin until we get HEAD. Suppose the total number of tosses is a random variable $X$, and $Pr(HEAD) = p$. What is the probability that $X$ is even? Denote this event ...
0
votes
0answers
22 views

Density of Gaussian Unitary Ensemble

I'm trying to learn a bit about Gaussian matrix ensembles, and am having some trouble making the following connection. Sorry if I'm being a bit obtuse. Take the Gaussian unitary ensemble (GUE) of $n ...
2
votes
0answers
80 views

Coupling Pairs of Random Variable.

Let $\{X_i\}_{i=1}^{n}$ and $\{Z_i\}_{i=1}^{n}$ be sets of independent random variables with coupling $\{X^{\hat{}}_i\}_{i=1}^{n}$, $\{Z^{\hat{}}_i\}_{i=1}^{n}$ respectively. It then states ...
1
vote
1answer
32 views

Random variable transformation function

I am stuck with a random variable transformation problem ($Y=\phi(X)$). The random variable $X$ has a uniform distribution $U(-1,1)$, and I want to transform it into $Y$ which is also an uniform ...
0
votes
0answers
15 views

Quantile function for multivariate random variable

Can anyone pls help me in resolving this issue? Let $w_k \sim \mathcal{N}(0,\mathbb{I}_{n\times n} )$ be a Gaussian mulativariate random variable, $p\in[0,1]$, $G\in \mathbb{R}_{r\times n}$, $H\in ...
0
votes
1answer
21 views

Mean of a PMF with a variable

I am given the following PMF: and I am asked to find the mean. I'm a bit lost on what to do with the ...
0
votes
1answer
27 views

Use of convolutions to compute the distribution of the sample mean?

Let's consider N i.i.d continuous random variables from some arbitrary distribution. Why do we have to approximate the distribution of the sample mean using the CLT? Why can't we explicitly compute ...
0
votes
1answer
18 views

Generate random number from set

You are given a set T of n non negative real numbers ${t_0, t_1 ... t_n}$ and probabilities $p_0,p_1...p_n$ where $\sum\limits_{i}^{n} p_1 = 1$. Assume $t_0 < t_1 ...< t_n$. Given a random ...
0
votes
0answers
18 views

Efficient algorithm for point estimation of a dependent random variable

Suppose $X$ is a normal-distributed random variable and $f$ is a known smooth function (possibly quite complicated, with many oscillations). Let $p(y)$ be the pdf of the dependent random variable $Y = ...
1
vote
1answer
28 views

Simplification of conditional expectation formula with two varibles

We are given a sequence of random varibles: $X_1,X_2,....X_n$ that each $X_i$ distribution is $Ber(p)$ We have : $N$ ~ $Poi( \lambda )$ We are also given : $S=X_1+X_2+....+X_n$ and: ...
1
vote
3answers
45 views

probability density function for a random variable

For a given probability density function $f(x)$, how do I find out the probability density function for say, $Y = x^2$? ...
0
votes
1answer
18 views

Expected value of $\max(X-Y,0)^2$ and $g(X_1,\ldots,X_n)$

Let $Y,Z$ be two continuous random variables with density functions $f_Y(y),f_Z(z)$ respectively. Then is it true that $$\mathbb E[(\max(Z-Y,0))^2] = \int_{-\infty}^\infty \int_{-\infty}^\infty ...
0
votes
1answer
43 views

probability of maximum of two independent random variable

Suppose $X$ and $Y$ are two independant random variable with exponential distribution with paramet $\lambda=1$ and $M=$max{$X$,$Y$}. Then $P(M \ge 4)$ is equal to : Answer: 0.036 how do i come to ...
2
votes
1answer
55 views

What's the distribution of the exponential of uniformly distributed variable?

I want to know the distribution of $z = \exp(j\varphi)$, with $\varphi \sim \mathcal{U}[-\pi;+\pi]$. From the book "Probability, Random Variables and Stochastic Processes" by Papoulis and Pillai I ...
0
votes
0answers
39 views

Bound on CDFs of sums of stochastically dominated random variables

Let M be a constant positive integer, and suppose that $X_1,\ldots X_M$ are non-negative dependent random variables. Suppose that $Y_1,\ldots Y_M$ are non-negative independent random variable where ...
1
vote
1answer
45 views

ruin of the gambler with probability to die

Consider a random walk on $\mathbb{Z}$ starting from $i >0$. With probability $p$ it moves to the nearest neighbor on the left, with the same probability it moves to the nearest neighbor on the ...
1
vote
2answers
61 views

Find P(X ≤ 2) and E(X). [closed]

Consider a game in which you rolls a single die until you accumulates a total of at least four dots. Let X denote the number of rolls needed. Find P(X ≤ 2) and E(X).
0
votes
2answers
99 views

When does equality in Markov's inequality occur?

Markov's inequality states that given any nonnegative random variable and $a>0$ then we have: $$P(X \geq a) \leq \frac{E(X)}{a}$$ At which $a$ is equality supposed to hold?