Mathematics.

probability distributions

Discrete Probability Distributions

Probability25 minDifficulty3 out of 10

You should know: random variables

Overview

A discrete probability distribution describes the probabilities of a random variable that can take only a countable number of values (finite or countably infinite), such as the outcomes of a die roll or the number of customer arrivals in an hour. It is characterized by a probability mass function (PMF) p(x) = P(X = x), which assigns a probability to each possible value, with all probabilities nonnegative and summing to 1. Common examples include the Bernoulli, binomial, Poisson, and geometric distributions, each modeling a different kind of discrete random process. The cumulative distribution function F(x) = P(X ≤ x) accumulates these probabilities and is a step function that jumps at each possible value of X.

Intuition

Imagine a vending machine that only dispenses whole items — you can't get 2.5 candy bars. A discrete distribution is the 'menu' of possible outcomes together with how likely each one is, like a bar chart where the heights of the bars (not areas, since each outcome is a single point) sum to exactly 1. Because outcomes are separate and countable, you can list them and their probabilities directly, unlike continuous distributions where you need a density and areas under a curve.

Formal Definition

Definition

A probability mass function p: X(Ω) → [0,1] must satisfy:

p(x)=P(X=x)≥0for all xp(x) = P(X = x) \ge 0 \quad \text{for all } x
Nonnegativity
∑xp(x)=1\sum_{x} p(x) = 1
Normalization
F(x)=P(X≤x)=∑k≤xp(k)F(x) = P(X \le x) = \sum_{k \le x} p(k)
Cumulative distribution function
E[X]=∑xx p(x)E[X] = \sum_x x\,p(x)
Expected value

Worked Examples

  1. Each of the 6 outcomes is equally likely.

    p(x)=16for x∈{1,2,3,4,5,6}p(x) = \tfrac{1}{6} \quad \text{for } x \in \{1,2,3,4,5,6\}

  2. Sum the probabilities across all outcomes.

    ∑x=16p(x)=6⋅16=1\sum_{x=1}^{6} p(x) = 6 \cdot \tfrac{1}{6} = 1

Answer: The PMF is uniform over {1,...,6} and correctly sums to 1.

Practice Problems

Difficulty 3/10

A random variable X has P(X=1)=0.3, P(X=2)=0.3, P(X=3)=p. Find p.

Difficulty 4/10

Using P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3, find F(1) = P(X ≤ 1).

Difficulty 5/10

A raffle sells 100 tickets; one wins $50 and the rest win $0. Let X be the winnings for a random ticket. Find E[X].

Quiz

A valid probability mass function must satisfy:
The cumulative distribution function F(x) of a discrete random variable is:
Which of these is an example of a discrete probability distribution?

Summary

  • A discrete distribution assigns probabilities via a PMF p(x) = P(X=x), with values nonnegative and summing to 1.
  • The CDF F(x) = P(X≤x) is a step function that accumulates PMF values.
  • Common discrete distributions include Bernoulli, binomial, Poisson, and geometric.

References

  1. WebsiteWikipedia — Probability distribution