Probability Mass Function (PMF)
What is PMF?
The PMF is a function that gives the probability of each possible discrete outcome for a random variable.
- Applies to: Discrete random variables
- Notation: \(P(X = x)\)
- Requirements:
- \(0 \leq P(X = x) \leq 1\)
- \(\sum_x P(X = x) = 1\)
Example: Fair Die Roll
A 6-sided die has possible outcomes \({1, 2, 3, 4, 5, 6}\). Since it is fair:
\(P(X = x) = \frac{1}{6} \quad \text{for } x = 1,2,3,4,5,6\)
This is a uniform PMF.
Interpretation
- Each bar in a PMF plot represents the exact probability of that outcome.
- No area under curve: Unlike PDFs, we do not interpret PMFs as densities or areas.
- Sum of bars must always equal 1: This is a sanity check for correctness.
Properties
| Property | Description |
|---|---|
| Discreteness | Only defined for countable outcomes (e.g., integers) |
| Additivity | Total probability over all outcomes is 1 |
| Visual Form | Bar graph showing probability for each outcome |
| Non-zero values | Only defined at discrete points; between values, probability is 0 |
Modified Example: Rigged Die
Let’s say the die cannot roll 3 or 4. Then a new PMF could be:
- \(P(X = 1) = 0.25\)
- \(P(X = 2) = 0.25\)
- \(P(X = 3) = 0\)
- \(P(X = 4) = 0\)
- \(P(X = 5) = 0.25\)
- \(P(X = 6) = 0.25\)
Sum still equals 1, but the distribution is non-uniform and has zero mass at 3 and 4.
Common Examples
- Bernoulli (1 trial, 2 outcomes):
\[
P(X = 1) = p,\quad P(X = 0) = 1 - p
\]
-
Binomial (n trials):
\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k} \] -
Geometric, Poisson, Hypergeometric: All have well-defined PMFs.
Mean and Variance
-
Expectation (Mean):
\[ \mathbb{E}[X] = \sum_x x \cdot P(X = x) \] -
Variance:
\[ \text{Var}(X) = \mathbb{E}[(X−μ)^2] = \sum_x (x - \mathbb{E}[X])^2 \cdot P(X = x) \]
Relationship to CDF
CDF accumulates the PMF up to a given value:
\[
F(x) = P(X \leq x) = \sum_{k \leq x} P(X = k)
\]
Example:
- \(F(4) = P(X \leq 4) = P(1) + P(2) + P(3) + P(4)\)
If 3 and 4 are rigged (0 probability), the CDF is flat between 2 and 5.