## The Central Limit Theorem

Robert Nishihara

[latexpage]The proof and intuition presented here come from this excellent writeup by Yuval Filmus, which in turn draws upon ideas in this book by Fumio Hiai and Denes Petz. Suppose that we have a sequence of real-valued random variables $$X_1, X_2, \ldots .$$ Define the random variable $$A_N = \frac{X_1 + \cdots + X_N}{\sqrt{N}}$$ to be a scaled sum of the first $N$ variables in the sequence. Now, we would like to make interesting statements about the sequence $$A_1, A_2, \ldots .$$

## A Geometric Intuition for Markov’s Inequality

[latexpage] As the title of the post suggests, this week I will discuss a geometric intuition for Markov’s inequality, which for a nonnegative random variable, $X$, states $$P(X \geq a) \leq E[X]/a.$$ This is a simple result in basic probability that still felt surprising every time I used it… until very recently. (Warning: Basic measure theoretic probability lies ahead. These notes look like they provide sufficient background if this post is confusing and you are sufficiently motivated!)