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Two probability distributions on (Borel subsets of) $\mathbb R$ are of the same "type" if for any random variable $X$ having one of those distributions, the other distribution is that of $\mu + \sigma …

$$ \begin{array}{cccccccccccc} & & \text{A} \\ & \swarrow & & \searrow \\ \text{B} & & & & \text{C} \\ & \searrow & & \swarrow \\ \downarrow & & \text{D} \\ & & & \searrow \\ \text{F} & & \rightarrow …

Kai Lai Chung once began a section of a textbook on probability by writing "Everybody knows" that $$ e^x = \sum_{n\,=\,0}^\infty \frac{x^n}{n!}. $$ (with those quotation marks). Other mathematicians …

Suppose $X$ is a random variable taking values in a finite set $\{a_1,\ldots, a_k \}$ and for $i=1,\ldots,k,$ $Y_i = \begin{cases} 1 & \text{if } X=a_i, \\ 0 & \text{otherwise.} \end{cases}$ \begin{a …

In independent Bernoulli trials with probability $p$ of success on each trial, let $X$ be the number of failures before the $n$th success. Then $$ \Pr(X=x) = \binom{-n}{\phantom{+}x} (-q)^x p^n \text{ …

The variance assigns a number to each of certain probability distributions on Borel subsets of $\mathbb R$. It has the properties of (1) shift-invariance, i.e. if $X$ is a random variable with a proba …

My actual question appears at the bottom of this posting. Suppose $$ \Pr(\Lambda\in d\ell) = \frac 1{\Gamma(\alpha)} \left( \frac\ell m\right)^{\alpha-1} e^{-\ell/m}\, \left( \frac{d\ell} m \right) \t …

Suppose you're a shopkeeper in the business of selling Items. An "Item" is a thing whose only property is that the quantity that can be bought by a purchaser must be a positive integer; all Items are …

The $n$th cumulant $\kappa_n$ of a probability distribution for $n\ge2$ is functional that is a polynomial in the first $n$ moments of the distribution, that has the properties of $(1)$ homogeneity, $ …

A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$ The mean of th …

This is getting no attention, so I'll try this here: The Robbins lemma, named after Herbert Robbins, says that if $X\sim\operatorname{Poisson}(\lambda)$ and $g$ is a function for which $\operatornam …

Once again stackexchange is not responding to one of my questions (so far no comments, no answers, two up-votes). Hence this "crossposting": Is there a simple (or not simple?) algorithm that will ch …

https://math.stackexchange.com/questions/64566/riemanns-zeta-function-and-the-uniform-distribution-on-1-0 Stackexchange isn't getting really excited about this, so here it is. The $n$th cumulant of …

Usually "exchangeable normal random variables" means jointly normal random variables $X_1,\ldots,X_n$ (i.e. so distributed that every linear combination of them is normally distributed) that are excha …

In a course taught by Morris Eaton on multivariate statistics that dealt mostly with the Wishart distribution, I learned this proposition: Suppose $$ M = \begin{bmatrix} A & B \\\\ B^T & C \end{bmatr …