# Tagged Questions

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### Probability of a single variable from a Moment Generating Function

This is from the A/S/M study guide, and the answer is listed, I just don't understand how he's arriving at the answer... I'm sure it's something simple I am missing! I have two identically ...
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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 ...
62 views

### 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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### Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i)$. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
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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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### Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
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### Difference between two independent geometric random variables

Let $\xi_1$ and $\xi_2$ be independent random variables: $\xi_1 \simeq Geom(1/2), \xi_2 \simeq Geom(1/6)$. How do you find the probability mass function of $\eta=\xi_1-\xi_2$ using convolution?
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### Analytic solution of the convolution of two discoutinous c.d.f s

I have a c.d.f of variable X with a mass point at the end point, $$F(x) = \begin{cases} 0 & x<a,\\ 1-\frac{m}{x+m-a} & a\le x < r-a,\\ 1 & x\ge r-a. \end{cases}$$ where m>0. Is it ...
377 views

### Finding distribution of $X^2+Y^2$ where $X,Y\sim N(0,1)$

Assume I have two random independent standard normal variables $X,Y\sim N(0,1)$, How can I find the distribution of $Z=X^2+Y^2$? I thought integrating the convolution, i.e ...
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### A continuous random walk of length 1

Suppose one starts at origo in in the plane and takes $N$ steps of length $1/N$ in a random direction, what is the distribution of the resulting distance from origo as $N$ approaches infinity? For one ...
1k views

### convolution of exponential distribution and uniform distribution

Given $X$ an exponentially distributed random variable with parameter $\lambda$ and $Y$ a uniformly distributed random variable between $-C$ and $C$. $X$ and $Y$ are independent. I'm supposed to ...
161 views

### What is the distribution of empirical covariance between two independent normal distributions?

Suppose that we have two independent normal distributions $\mathcal{N}_{1}(0,s)$, $\mathcal{N}_{2}(0,t)$ what is the distribution of empirical covariance (or empirical correlation if this make my ...
328 views

### convolution square root of uniform distribution

I need to find a probability distribution function $f(x)$ such that the convolution $f * f$ is the uniform distribution (between $x=0$ and $x=1$). I would like to generate pairs of numbers with ...
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### what's the distribution of the inverse of a random variable that follows a negative binomial distribution?

I was studying the method of moments estimation of parameters, and I encountered the following problem. I have a geometric distribution as following: $P(X=k) = p(1-p)^{k-1}$, and a sample size of n, ...
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### Probability density of vector sum

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 ...
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### Convolution of two functions (pdfs)

I want to convolve two signals . The range of each of the signal is 0 to 1. ...
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### Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
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### Computing convolution density function with Maple

I would like to know how to compute a probability function of a convolution of Negative Binomial distribution with Maple. Here is an easy example of what I want to do : ' ...
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### The distribution of the sum of two independent exponential distributions

I am trying to calculate the distribution of the sum of two independent log-uniform distributions but something doesn't add up. Suppose $a \sim \mathrm{uni}(0,1)$ and $b \sim \mathrm{uni}(0,1)$. ...
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### Derivation of pmf from convolution

Suppose that a discrete random variable (with finite support) $Y$ is given by $Y = X_1 - X_2$, where $X_1$ and $X_2$ are both discrete random variables with finite support and with the same ...