# Tagged Questions

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### Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
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### Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
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### Question about transformations and sums on uniformly distributed random variables.

I'm looking into a few problems as a hobby of mine, and found myself with the following problem: let $X$ be a random variable uniformly distributed on $[0,1]$. What is the probability that after $N$ ...
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### What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
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### Pdf of $Z=(XY)^{1/2}$. with X,Y independent r.v. with the same distribution (iid) [closed]

Let be $X,Y$ two independent random variables having the same distribution (the following is the density of this distribution) $$f(t)= \frac{1}{t^2} \,\,\, \text{for t>1}$$ Calculate the ...
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### What's the density of $Z=\max(X,Y)-\min(X,Y)$ with $X,Y$ exponentials of parameter $\lambda$?

Let be $X,Y$ two independent exponential random variables with parameter $\lambda$. What is the pdf of $Z=\max(X,Y)-\min(X,Y)$? Thanks for your help.
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### Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
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### Convolution of exponential family distributions

Is there a general form for the convolution of two exponential family distributions?
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### Sum of two independent random variable, Convolution.

I need the graphic of this two function to evaluate this correlation?
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### Convolution of finite measures

I am puzzled by the following (maybe very stupid) question I stumble upon in the course of a project: let $p$ be a probability measure on some abelian group $E$ (actually, $E=\mathbb{Z}_n$ with its ...
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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 ...
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### Is it possible to obtain the Uniform distribution as the difference of two independent random variables?

Is it possible to have two independent random variables X,Y with identical distribution, such that $X-Y \sim \text{Uniform}[a,b]$? I am almost certain that is not, but maybe I am overlooking ...
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### How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
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### A question about stochastic ordering and convolution

Two probability density functions $f$ and $g$ are known to have distribution functions $F$ and $G$ respectively with $F(y)>G(y)$ for all $y$, say on $\mathbb{R}$. It is known that if we convolve ...
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### Adding truncated normals: calculating convolutions

Problem: Suppose that $X$, $Y$, and $Z$ are independent standard normal random variables. What is the probability of: P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, ...
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### Confused with estimator for random variables.

I am working on a practice exercise in preparation for a final this week. I am really stuck on the following problem: Let $X_1, X_2$ be a random sample for a population with the probability density ...
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### Fourier transform of product

I would like to know the fourier transform of the product of the Cauchy probability distribution $f(x)=\frac{1}{\pi (1+x^2)}, -\infty<x<\infty$ with itself. I know that the fourier transform of ...
Consider two unit $\mathbb R^2$ vectors $v$ and $w$. Then $v+w$ lies within a (closed) circle with radius 2, that is, in the region $x^2+y^2\leq4$. Intuitively, the probability of $v+w$ lying close ...