# Tagged Questions

Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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### Proof that the sum of two Gaussian variables is another Gaussian

The sum of two Gaussian variables is another Gaussian. It seems natural, but I could not find a proof using Google. What's a short way to prove this? Thanks! Edit: Provided the two variables are ...
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### Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
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### If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.

Knowing that if you have two independent $X$ and $Y$, and $f$ and $g$ measurable functions, how to show that then $U = f (X)$ and $V = g (Y)$ are still independent.
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### Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $(\Omega, \mathscr{A}, \mu )$ ...
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### Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n$ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
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### Sum of independent Gamma distributions is a Gamma distribution

If $X\sim \mathrm{Gamma}(a_1,b)$ and $Y \sim \mathrm{Gamma}(a_2,b)$, I need to prove $X+Y\sim(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and ...
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### How to compute the sum of random variables of geometric distribution

Let $X_{i}$, $i=1,2,\dots, n$, be independent random variables of geometric distribution, that is, $P(X_{i}=m)=p(1-p)^{m-1}$. How to compute the PDF of their sum $\sum_{i=1}^{n}X_{i}$? I know ...
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### probability distribution of coverage of a set after X independently, randomly selected members of the set

I have a set of numbers where I am randomly and independently selecting elements within a set . After a number of these random element selections I want to know the coverage of the elements in the ...
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### mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
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### The joint density of the max and min of two independent exponentials

Let $X=\min(S,T)$ and $Y=\max(S,T)$ for independent exponential variables $S$ and $T$. Find the joint density of $X$ and $Y$. Are $X$ and $Y$ independent? How would you suggest I approach this?
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### How To prove Any Change to $v=a\cdot y + b$ maks $y=(a)^{-1}(v-b)$ Uni. random value

This question may seem to be related to Probability and Data Integrity but mine is much simpler and consideres a DIFFERENT problem. Let a finite field be $\mathbb{Z}_p$, where $p$ is a prime number. ...
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### $X,Y$ are independent exponentially distributed: What is the distribution of $X/(X+Y)$ [duplicate]

I'm a business major, so really distribution theory is not my strong suit. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything ...
$Y$ is a random variable with $$M(t) = \frac{1}{(2-\exp(t))^s}.$$ Does $$\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$$ converge in distribution as $s$ tends to infinity? I let $Z = \frac{Y-E(Y)}{\... 3answers 5k views ### Mean distance between 2 points within a sphere I have found an answer on this site to the question of determining the mean straight-line distance between 2 randomly chosen points in a disc of radius r. I'm now trying to find an answer to the same ... 1answer 599 views ### Question about order statistics I saw a paper which says that: Let$Z_i$be i.i.d. exponential random variables with mean$1$, and let$S_n = Z_1 + \dots + Z_n$for all$n$. For a fixed$n$, let$U_j = S_j/S_{n+1}$, then$(U_1,...
How to calculate the following probability $P(X \leq x, Y=y)$ where $X$ is a continuous random variable and $Y$ is a discrete random variable. I have been given the distribution of $X$ and ...