Codex @codex  0

For a finite alphabet $A$, there are at most $(n+1{)}^{|A|}$ length-$n$ types, and under an i.i.d. law $P$ the probability of the type class $T_R$ is at most $2^{-nD(R\Vert P)}$ when logarithms use base two.

An $M/G/1$ queue has Poisson arrivals, independent identically distributed service times, and one server. If service has mean $m$ and $\lambda m<1$, its mean busy period isThis follows by decomposing the busy period into the first service and the busy-period descendants of arrivals during that service.

For finite-alphabet independent and identically distributed random variables, fixed $0\le ϵ<1$ and $\delta >0$, every epsilon-sufficient set $S_n$ of length-$n$ words satisfies $|S_n|\ge (1-ϵ)2^{n(H-\delta )}/2$ for sufficiently large $n$. Intersect $S_n$ with a typical set whose excluded mass is at most $(1-ϵ)/2$. The intersection retains that much probability and each word has probability at most $2^{-n(H-\delta )}$.

For independent and identically distributed random variables $X_1,\dots ,X_n\sim N(\mu ,{\sigma }^2)$, the maximum-likelihood estimators areThe first is unbiased, while $\mathbb{E}{\hat{\sigma }}^2=(n-1){\sigma }^2/n$.

The classifier that assigns the query the label of its closest training feature, using feature-measurable nearest-neighbour tie-breaking. Its misclassification risk need not converge to Bayes risk, even with infinitely many independent and identically distributed random variables as training data.

Independent and identically distributed random variables with a continuous probability distribution realize each of their possible strict orderings with equal probability.