Premed · Premed · Statistics Biostatistics
Lecture 8: Discrete Probability Distributions
Statistics / Biostatistics
Learning Objectives
By the end of this lecture, students will be able to:
- Define a random variable and distinguish discrete from continuous random variables
- Construct and interpret a probability distribution for a discrete random variable
- Calculate the expected value (mean) and variance of a discrete distribution
- Identify the properties that all valid probability distributions must satisfy
- Apply discrete distributions to clinical examples
Lecture Content
I. Random Variables
A random variable is a numerical quantity whose value is determined by the outcome of a random experiment. Random variables are denoted by uppercase letters (X, Y, Z), while specific values they take are denoted by lowercase letters (x, y, z). A discrete random variable takes on a countable number of distinct values, such as the number of patients admitted per day, the number of adverse events, or the number of positive tests in a batch. A continuous random variable can take any value in an interval or collection of intervals, such as blood glucose level, time to event, or weight. Continuous random variables are covered in subsequent lectures.
II. Probability Distribution of a Discrete Random Variable
A probability distribution is a table, formula, or graph that lists all possible values of X and their associated probabilities. For a distribution to be valid, two requirements must be met: P(X = x) must be greater than or equal to zero for all x, and the sum of all P(X = x) must equal 1. The function f(x) = P(X = x) is called the probability mass function (PMF).
As an example, let X represent the number of side effects experienced by a patient, with possible values 0, 1, 2, and 3. If P(X=0) = 0.40, P(X=1) = 0.30, P(X=2) = 0.20, and P(X=3) = 0.10, this constitutes a valid probability distribution because 0.40 + 0.30 + 0.20 + 0.10 = 1.00 and all probabilities are non-negative.
III. Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF), denoted F(x) = P(X <= x), gives the probability that X takes a value less than or equal to x. It equals the sum of P(X = k) for all k <= x. For discrete variables, the CDF is a non-decreasing step function. It is useful for calculating probabilities over ranges: P(X > 1) = 1 - F(1) = 1 - P(X <= 1), and P(1 <= X <= 3) = F(3) - F(0).
<image>A two-panel figure for a discrete random variable (number of side effects, 0-3). Panel A: A bar chart showing the probability mass function (PMF), with bars at x = 0, 1, 2, 3 and heights 0.40, 0.30, 0.20, 0.10. Panel B: A step function showing the cumulative distribution function (CDF), starting at 0 and stepping up to 0.40, 0.70, 0.90, and 1.00 at the corresponding x values. Both panels have clearly labeled axes.</image>
IV. Expected Value (Mean) of a Discrete Random Variable
The expected value of a discrete random variable is defined as E(X) = mu = sum of [x * P(X = x)] for all possible values of x. It represents the long-run average value of X over many repetitions and does not need to be a value that X can actually take.
Using the side effects distribution from earlier, E(X) = 0(0.40) + 1(0.30) + 2(0.20) + 3(0.10) = 0 + 0.30 + 0.40 + 0.30 = 1.00. On average, patients experience 1.0 side effects. Two important properties of expected value are that E(aX + b) = a * E(X) + b, and E(X + Y) = E(X) + E(Y), which holds regardless of whether X and Y are independent.
V. Variance and Standard Deviation
The variance of a discrete random variable is Var(X) = sigma^2 = sum of [(x - mu)^2 * P(X = x)], with the equivalent computational formula Var(X) = E(X^2) - [E(X)]^2. The standard deviation is sigma = sqrt(Var(X)).
Continuing the example, E(X^2) = 0^2(0.40) + 1^2(0.30) + 2^2(0.20) + 3^2(0.10) = 0 + 0.30 + 0.80 + 0.90 = 2.00. Therefore Var(X) = 2.00 - (1.00)^2 = 1.00, and SD(X) = 1.00. Two key properties of variance are Var(aX + b) = a^2 * Var(X), and if X and Y are independent, Var(X + Y) = Var(X) + Var(Y).
VI. Bernoulli Distribution
The Bernoulli distribution is the simplest discrete distribution, modeling a single trial with exactly two outcomes. X = 1 (success) with probability p, and X = 0 (failure) with probability 1 - p. The PMF is P(X = x) = p^x * (1 - p)^(1-x) for x = 0, 1. The expected value is E(X) = p, and the variance is Var(X) = p(1 - p). A clinical example would be a single patient either responding to treatment (X=1) or not (X=0). The Bernoulli distribution is the building block for the binomial distribution, which models multiple independent Bernoulli trials.
VII. Uniform Discrete Distribution
In a uniform discrete distribution, all outcomes are equally likely. If X takes values 1, 2, ..., n, then P(X = k) = 1/n for each k. The expected value is E(X) = (n + 1) / 2, and the variance is Var(X) = (n^2 - 1) / 12. A familiar example is rolling a fair six-sided die, where each face has a probability of 1/6.
VIII. Linear Combinations of Random Variables
When working with random variables X and Y that have known means and variances, the expected value of a linear combination is E(aX + bY) = a E(X) + b E(Y). If X and Y are independent, the variance of the linear combination is Var(aX + bY) = a^2 Var(X) + b^2 Var(Y). These rules are important for combining measurements, creating composite scores, or analyzing differences between groups.
<image>A summary comparison table of the Bernoulli and discrete uniform distributions. For each: the PMF formula, a small bar chart visualization, the expected value formula, and the variance formula. A clinical example accompanies each: Bernoulli (patient responds or not to treatment, p = 0.7) and Uniform (random assignment to one of 4 treatment arms, each with probability 0.25). The bar charts visually show the probability distributions.</image>
IX. Applications in Clinical Biostatistics
Discrete probability distributions have numerous clinical applications. They are used to model the number of patients who respond to a therapy in a clinical trial, to predict the expected number of disease cases in a surveillance sample, and to calculate probabilities for quality control such as the number of defective items in a batch of medical devices. Risk assessment relies on expected values to estimate the expected number of adverse events per patient. Understanding expected value is also foundational for cost-effectiveness analysis, where the expected cost is calculated as the sum of each outcome's cost multiplied by its probability.
