$\newcommand{\R}{\mathbb R}\newcommand{\e}{\varepsilon}\newcommand{\0}{\mathbf 0}\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}$Consider a linear regression model $y = X\beta+\e$ with $X\in\R^{n\times p}$ known and full column rank. Under common…
Yesterday at the office we played a rousing game of holiday trivia, and one question asked for the total number…
$\newcommand{\d}{\mathbf d}\newcommand{\one}{\mathbf 1}\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}$In this post I’m going to look at the degree distribution of stochastic block models. $n$ will denote…
$\newcommand{\X}{\mathcal X}\newcommand{\x}{\mathbf x}\newcommand{\0}{\mathbf 0}\newcommand{\one}{\mathbf 1}\newcommand{\y}{\mathbf y}\newcommand{\z}{\mathbf z}$In this post I’m going to look at distributions over polynomials using Gaussian processes.…
$\newcommand{\R}{\mathcal R}\newcommand{\C}{\mathcal C}\newcommand{\Rbb}{\mathbb R}\newcommand{\0}{\mathbf 0}\newcommand{\one}{\mathbb 1}\newcommand{\span}{\operatorname{span}}\newcommand{\rank}{\operatorname{rank}}$In this post I’m going to prove the Steinitz exchange lemma, that the row rank…
$\newcommand{\Om}{\Omega}\newcommand{\F}{\mathscr F}\newcommand{\one}{\mathbf 1}\newcommand{\R}{\mathbb R}\newcommand{\e}{\varepsilon}\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}\newcommand{\convas}{\stackrel{\text{a.s.}}{\to}}\newcommand{\w}{\omega}\newcommand{\N}{\mathbb N}\newcommand{\convp}{\stackrel{\text{p}}{\to}}$In this post I’m going to introduce almost sure convergence for sequences of random variables, compare…
$\newcommand{\E}{\operatorname{E}}\newcommand{\Xmin}{X_{(1)}}\newcommand{\Fmin}{F_{(1)}}\newcommand{\fmin}{f_{(1)}}\newcommand{\eff}{\operatorname{eff}}$In this post I’m going to look at two probability questions that arose in the context of a friendly game…
$\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}$In this post I’m going to evaluate $$ \lim_{n\to\infty} (n!)^{1/n} $$ by connecting it to the harmonic series.…
$\newcommand{\Om}{\Omega}\newcommand{\w}{\omega}\newcommand{\F}{\mathscr F}\newcommand{\R}{\mathbb R}\newcommand{\e}{\varepsilon}\newcommand{\convd}{\stackrel{\text d}\to}\newcommand{\convp}{\stackrel{\text p}\to}\newcommand{\convas}{\stackrel{\text {a.s.}}\to}\newcommand{\E}{\operatorname{E}}\newcommand{\Var}{\operatorname{Var}}\newcommand{\one}{\mathbf 1}$In this post I’m going to introduce two modes of convergence for random variables.…
$\newcommand{\X}{\mathcal X}\newcommand{\one}{\mathbf 1}$In this post I’m going to look at the following: suppose I have a random variable $X$ supported…