Random Variables and Distributions

A random variable assigns a numeric value to each outcome of a random experiment. We don't care about the mechanism producing the value — we care about the distribution over its possible values.

Discrete vs continuous

A discrete random variable takes on values from a countable set (the result of a die roll, the number of customers in a queue). A continuous random variable can take any value in a range (height, temperature). The distribution is described by a probability mass function (PMF) in the discrete case and a probability density function (PDF) in the continuous case.

Common discrete distributions

• Bernoulli($p$): 1 with probability $p$, 0 with probability $1-p$. A single coin flip.
• Binomial($n, p$): number of successes in $n$ independent Bernoulli trials.
• Poisson($\lambda$): count of events in a fixed interval when events happen at average rate $\lambda$.
• Geometric($p$): number of trials until the first success.

Common continuous distributions

• Uniform($a, b$): all values in $[a, b]$ equally likely.
• Normal($\mu, \sigma^2$): the Gaussian "bell curve" — central limit theorem makes it appear everywhere.
• Exponential($\lambda$): time between events in a Poisson process; memoryless.
• Beta($\alpha, \beta$): distribution over probabilities; the conjugate prior for the Bernoulli.

CDF

The cumulative distribution function $F(x) = P(X \leq x)$ works for both discrete and continuous. It's always non-decreasing, starts at 0 ($x \to -\infty$), ends at 1 ($x \to \infty$). $F$ uniquely determines the distribution.

Why this matters

Every model in statistics and machine learning assumes some distribution — usually as a prior, a likelihood, or a noise term. Knowing the standard distributions and when they apply lets you read papers, debug models, and reason about which assumption you're violating.