Probability Mass Function

Definition

A probability mass function assigns probabilities to individual discrete values.

Short form:

PMF

Example:

X = die roll

PMF:

P(X = 1) = 1/6
P(X = 2) = 1/6
P(X = 3) = 1/6
P(X = 4) = 1/6
P(X = 5) = 1/6
P(X = 6) = 1/6

PMF = probability table for discrete outcomes.

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Why Mass?

The word "mass" means probability is placed on specific values.

For a discrete random variable, each possible value can receive its own probability.

Example:

P(X = 3) = 1/6

The value 3 itself has probability mass.

Discrete probability can sit directly on individual values.

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Valid PMF

A valid PMF must satisfy:

each probability >= 0
all probabilities add to 1

Example:

0.2 + 0.3 + 0.5 = 1.0

Invalid:

0.2 + 0.3 + 0.7 = 1.2

The total probability cannot exceed 1.

A PMF must distribute all probability across possible discrete values.

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PMF As A Function

A PMF can be written as a function.

Example:

p(x) = P(X = x)

For a fair die:

p(1) = 1/6
p(2) = 1/6
...
p(6) = 1/6

The function takes a possible value and returns its probability.

A PMF maps values to probabilities.

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PMF vs Distribution

A probability distribution is the general idea.

A PMF is one way to represent a distribution for discrete variables.

distribution -> general structure of uncertainty
PMF -> discrete distribution as value-probability assignments

A PMF is a discrete probability distribution.

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PMF In Classification

For classification, labels are discrete.

Example:

Y = category

Possible values:

bug
feature
question

A model can output:

P(Y = bug) = 0.7
P(Y = feature) = 0.2
P(Y = question) = 0.1

This behaves like a PMF over labels.

Classification probabilities form a distribution over discrete classes.

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Final Summary

• A PMF assigns probabilities to discrete values.
• Each probability must be nonnegative.
• All probabilities must add to 1.
• A PMF maps a possible value to its probability.
• PMF is the discrete form of a probability distribution.
• Classification outputs can be viewed as PMFs over labels.