Random variables

Discrete and continuous variables and their distributions.

A random variable $X$ maps outcomes to numbers. A discrete random variable takes countable values and has a probability mass function (PMF) $p (x) = P (X = x)$ with $\sum_{x} p (x) = 1$ . A continuous one has a probability density function (PDF) $p (x)$ with $\int p (x) d x = 1$ ; probabilities are areas, $P (a \leq X \leq b) = \int_{a}^{b} p (x) d x$ .

The cumulative distribution function $F (x) = P (X \leq x)$ works for both kinds and rises from 0 to 1.

Common distributions: Bernoulli (one coin flip, parameter $p$ ), binomial (number of heads in $n$ flips), categorical (one roll of a $K$ -sided die), uniform, and the Gaussian (normal) $N (μ, σ^{2})$ :

p (x) = \frac{1}{2 π σ ^{2}} exp (- \frac{( x - μ ) ^{2}}{2 σ ^{2}})

Formula to remember

Bayes' theorem

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

Posterior = likelihood × prior / evidence.

Quick check · +5 XP

What is the Euclidean distance between A(3, 5) and B(6, 9)?

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