# Tagged Questions

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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Help with random variable to found probabilty (PDF)

Stuck in this example to found (PDF) in many conditions
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $X$ ?
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### 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 ...
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### 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, ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...