Random Variables

A random variable $X$ is a function from outcomes to numbers, equipped with a probability distribution describing the likelihood of each value.

Discrete RVs:

• PMF (probability mass function): $p(x) = P(X = x)$, must satisfy $\sum_x p(x) = 1$
• Example: coin flip, dice roll, word count

Continuous RVs:

• PDF (probability density function): $f(x)$ where $P(a \leq X \leq b) = \int_a^b f(x)dx$
• $f(x) \geq 0$ and $\int_{-\infty}^{\infty} f(x)dx = 1$
• Note: $f(x)$ can exceed 1; it's a density, not a probability

CDF (cumulative distribution function): $F(x) = P(X \leq x)$

• Works for both discrete and continuous
• Non-decreasing, right-continuous, $F(-\infty) = 0$, $F(\infty) = 1$
• For continuous RVs: $f(x) = F'(x)$

Joint distributions: $p(x, y)$ for two RVs. Marginal: $p(x) = \sum_y p(x,y)$. Conditional: $p(x|y) = p(x,y)/p(y)$.

Independence: $X \perp Y$ iff $p(x,y) = p(x)p(y)$ for all $x, y$.

See also: Expectation and Variance, Key Probability Distributions, Bayes' Theorem