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Some general remarks. There is a family of linear operators $\phi_{a,d}$ which, given a generating function $f(x) = \sum f_n x^n$, returns the generating function $\sum f_{an+d} x^n$. Observe that, …

The distribution you're looking at is called the binomial distribution.

Yes. Here's what should be a proof: the set of pairs of elements satisfying any particular relation is Zariski closed, hence has measure zero (to show that it is not $SO(3) \times SO(3)$ it suffices …

More or less, yes.

No. Let's restrict our attention to $\mathbb{N}$. The hypotheses imply that if $q$ is a prime, then the probability that a random positive integer is not divisible by $q$ is $1 - \frac{1}{q}$. They al …

In his notes on free probability, Terence Tao describes a general approach to non-commutative probability which prioritizes the algebra of random variables above the sample space; I find this conceptu …

No. Because $c_n$ is quadratic, the values of $m_0, m_1, m_2$ can be arbitrary (subject to $m_0 = 1$ since presumably we are looking at a probability distribution). The first condition that needs to b …

People are actively working on this, although maybe not many people. See, for example, Uniform coherence by Garrabrant, Fallenstein, Demski, and Soares, and the references therein. The abstract: W …

Here are examples showing that unlike in the previous problem, here it does not suffice to simply use the fact that the harmonic series / the sum of the reciprocals of the primes diverges. In fact for …

This isn't an answer to your question, but without any restrictions, the expected number of $r$-cycles in a permutation of at least $r$ elements is $\frac{1}{r}$, so the expected number of cycles in a …

This may not be quite what you had in mind, but: suppose you were trying to compute the absolute value of a Gauss sum $\sum_{a=0}^{p-1} \zeta^{a^2}$ where $\zeta = e^{ \frac{2 \pi i}{p} }, p$ an odd p …

Equivalently, we are considering a random function $f : [n] \to [n]$ where $[n] = \{ 1, 2, \dots n \}$ is a finite set of size $n$, which assigns to each prank cigarette a pack. The second question is …

Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows: Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2 …

Suppose I toss $n$ coins. It's natural to model this probabilistically in terms of a sample space $\{ H, T \}^n$ constructed as the product of $n$ copies of the sample space of possible outcomes of a …

One answer I've heard is that you want the notion of standard deviation to 1) apply to points in Euclidean space, and 2) to be invariant under rotation. You don't get the second property unless you s …