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For $i, j \in \{ 1, \ldots, n \}$, let $X_{i,j}$ be a real-valued random variable uniformly distributed on the interval $[0,1]$. The $X_{i,j}$ are independent. Let $A_{i,j}$ be the indicator random …

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 mo …

Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x. A couple weeks ago in a technical s …

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What …

While playing around with this question (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is …

I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both asymptotical …