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Let $p$ be a positive integer (which is not a power of $2$), and suppose we want to generate a number uniformly randomly in the set $\{ 0, 1, \dots , p-1 \}$ (to emulate a dice roll). We are given acc …

If $r < \frac{1}{3}$, then the graph cannot possibly be connected. Indeed, the largest possible connected components are of four points clustered around one of the lattice points. Proof: Suppose we h …

This is by no means a complete classification, although I'm surprised that no-one has mentioned that we can easily construct the most well-known continuous distributions from rational functions of $N( …

Yes, your conjecture is true. Suppose otherwise. Then there exists a counterexample $f : \mathcal{P}(8) \rightarrow \{0, 1\}$. For each set $X \in \mathcal{P}(8)$, let the proposition $P_X$ denote $f …

The integrand is periodic modulo $\pi$ in each variable, so it suffices to integrate each variable over $[0, \pi]$ and replace the constant factor by $2^{-7}$. If we were to apply a change of variabl …