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Also known as: PMF
A function that gives the probability that a discrete random variable equals a specific value.
DESCRIPTION
A probability mass function (PMF) assigns probabilities to each value in the support of a discrete random variable. PMFs sum to one across all possible outcomes. They are fundamental in modeling counts and categorical phenomena such as clicks, defects, and arrivals. Common PMFs include Bernoulli, binomial, geometric, and Poisson distributions. PMFs enable calculation of expectations, variances, and event likelihoods via summations, and connect to the cumulative distribution function through discrete sums.
ALTERNATIVE DEFINITIONS
Wolfram MathWorld
For a discrete random variable X, the probability mass function p(x) gives the probability that X takes the value x. The sum over all x equals 1.
States summation normalization for discrete variables.
StatLect
A probability mass function assigns a probability to each possible value of a discrete random variable and the sum of all assigned probabilities equals one.
Focuses on discrete support and summation to one.
NIST/SEMATECH e-Handbook of Statistical Methods
For discrete data, the probability mass function (pmf) gives the probability that the discrete random variable equals a specific value and probabilities sum to one.
Connects discrete nature to normalization explicitly.
RELATED TERMS
Cumulative Distribution Function Random Variable Discrete Distribution Bernoulli Distribution Binomial Distribution Poisson Distribution Mean Support Likelihood Function
COMPETING TERMS
RELATIONS
- Narrower: Categorical Distribution
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist AI Engineer Data Analyst MLOps Engineer
EXTERNAL SOURCES
- Wikipedia - Probability mass function
- NIST/SEMATECH e-Handbook - Discrete distributions and PMFs
- SciPy Stats - SciPy PMF functions and usage
- Wikibooks - Open textbook on distributions
TOP QUESTIONS
What does a Probability Mass Function describe?
It describes the probability that a discrete random variable takes a particular value, assigning probabilities to each outcome in its support.
Category: Definition
How does a Probability Mass Function differ from a Probability Density Function?
A PMF uses sums for discrete values; a PDF uses integrals over intervals for continuous variables.
Category: Comparison
Why must PMF probabilities sum to one?
The normalization ensures total probability across all possible discrete outcomes equals one, satisfying probability axioms.
Category: Requirements
What are common examples of PMFs?
Bernoulli, binomial, geometric, and Poisson distributions frequently model counts, successes, and arrivals.
Category: Overview
How do you compute expected values using a PMF?
Take the sum over all values of x multiplied by p(x), yielding the mean of the distribution.
Category: How to
Can a PMF assign probability to values outside the support?
No, probabilities are zero for values not in the discrete support, ensuring consistency of the model.
Category: Misunderstanding
How is a CDF derived from a PMF?
The CDF at x is the sum of the PMF over all values less than or equal to x.
Category: Concept
Where is a PMF used in machine learning?
Classification likelihoods, count models, Naive Bayes with discrete features, and emission models in HMMs use PMFs.
Category: Application
How can you estimate a PMF from data?
Use empirical frequencies, maximum likelihood for parametric families, or Bayesian estimation with appropriate priors.
Category: How to