Discrete Probability Distributions

A rigorous guide to probability mass functions, cumulative probabilities, expectation, variance, and model checks.

A probability distribution pairs each outcome with a nonnegative weight totalling one. Discrete outcomes have probability bars whose heights sum to one. A weighted balance marker locates expected value, which need not itself be a possible outcome.

Precise definition

A discrete random variable $X$ has a probability mass function $p(x)=P(X=x)$ on a finite or countable support, with $p(x)\ge0$ and $\sum_xp(x)=1$. Its cumulative distribution is $F(x)=P(X\le x)=\sum_{t\le x}p(t)$.

Notation and mathematical language

The expectation is $E[X]=\sum xp(x)$ when absolutely convergent; variance is $E[(X-\mu)^2]=E[X^2]-\mu^2$. Standard deviation is the nonnegative square root. Expectation is a long-run or probability-weighted centre, not necessarily a possible outcome.

Conceptual picture

A mass function places probability at individual support points. Bar heights are probabilities, while cumulative values add all mass to the left. Transformations use $E[g(X)]=\sum g(x)p(x)$ without needing the distribution of $g(X)$ first.

Fully worked example

Interpretation and application

Discrete distributions model counts, claims, arrivals, and games. Expected profit summarizes repeated risk under a model; it does not guarantee any one outcome or capture risk preference by itself.