Probability mass function
A probability mass function (PMF) assigns to each possible value of a discrete random variable the exact probability that the variable equals that value. All probabilities are non-negative and must sum to exactly one.
See it move
In a batch of three items, each independently defective with probability 0.1, the number of defects X follows a binomial PMF. The probability of zero defects is 0.729, of one defect 0.243, of two defects 0.027, and of three defects 0.001. These four values are all non-negative and add up to exactly one, covering every possible outcome for X.
The formula
Variables
- probability mass at value x (dimensionless)
- discrete random variable (dimensionless)
- specific value in the support of X (dimensionless)
Variables
- probability mass at x (dimensionless)
Validity conditions for a PMF: p(x) ≥ 0 for all x in the support, and the probabilities sum to exactly one.
Check yourself
A discrete random variable X has PMF: P(X = 0) = 0.3, P(X = 1) = 0.5, P(X = 2) = 0.2. A student claims P(X = 1.5) = (0.5 + 0.2) / 2 = 0.35, interpolating between adjacent support values. What is the fundamental error in this reasoning?