Probability Mass Function (PMF)

Deep Breakdown

What it tells me (Situation Awareness): The PMF assigns a probability to each specific value that a discrete random variable can take. It is the complete specification of a discrete distribution. In DA/DS: used for Bernoulli, Binomial, Poisson, and Negative Binomial distributions that model counts, binary outcomes, and rare events.

Flowchart

Real-world use

Where it is used

Modeling the number of support tickets per day (Poisson), probability of conversion (Bernoulli), number of successes in $n$ trials (Binomial), or count data in general.

Problem it solves

Provides an exact probability for each discrete outcome — unlike continuous distributions where $P(X = x) = 0$.

How it solves the problem

Maps each possible discrete value to a non-negative probability, where all probabilities sum to 1.

Example in a business

A/B test with 1000 users: $X \sim \text{Binomial}(n=1000, p=0.05)$. $P(X = 50) = \binom{1000}{50} (0.05)^{50} (0.95)^{950}$. Used to assess how likely exactly 50 conversions are under the null.

The formula, piece by piece

$f(x) = P(X = x)$

• $f(x)$: the PMF at value $x$.
• $P(X = x)$: probability that the random variable $X$ equals exactly $x$.
• Must satisfy: $f(x) \geq 0$ for all $x$, and $\sum_x f(x) = 1$.

The formula in words

"The PMF at $x$ equals the probability that the random variable $X$ takes exactly the value $x$."

Critical Concepts

• Discrete only: PMF assigns exact probabilities. CRVs use PDF instead.
• CDF from PMF: $F(x) = \sum_{t \leq x} f(t)$.
• PMF bar chart: each bar height = exact probability.
• Python: scipy.stats.<dist>.pmf(k, ...).

Step-by-step calculation (tiny example)

Step 1: Fair die $X$ = outcome of a fair 6-sided die. $f(x) = 1/6$ for $x \in \{1,2,3,4,5,6\}$.

Step 2: Verify $\sum_{x=1}^{6} f(x) = 6 \times (1/6) = 1$. Valid PMF.

Step 3: Query $P(X \leq 3) = f(1) + f(2) + f(3) = 3/6 = 0.5$.

Code examples

Beginner example (Python)

from scipy.stats import binom   # PMF for Binomial: n=10 trials, p=0.3 success rate
n_trials = 10
p_success = 0.3
k_values = range(n_trials + 1)
pmf_values = [binom.pmf(k, n_trials, p_success) for k in k_values]
print(f"{'k':>4} {'P(X=k)':>10}")
for k, p in zip(k_values, pmf_values):
    print(f"{k:>4} {p:>10.6f}")

Beginner example (R)

n_trials <- 10; p_success <- 0.3
k_values <- 0:n_trials
pmf_values <- dbinom(k_values, n_trials, p_success)
for (i in seq_along(k_values)) { cat(sprintf("P(X=%2d) = %.6f\n", k_values[i], pmf_values[i]))
}

Professional example (Python)

import numpy as np
from scipy.stats import binom, poisson
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

def compare_pmfs( n_trials: int, p_success: float, lambda_rate: float | None = None ) -> dict[str, np.ndarray]:
    """Compare Binomial and Poisson PMFs for count modeling. When lambda = n*p (large n, small p), Poisson approximates Binomial. Args: n_trials: Binomial n parameter. p_success: Binomial p parameter. lambda_rate: Poisson lambda (defaults to n*p). Returns: Dict with k values and PMFs for both distributions. """
    lam = lambda_rate if lambda_rate is not None else n_trials * p_success
    k_range = np.arange(0, min(int(lam * 5) + 1, n_trials + 1))
    return { "k": k_range, "binomial_pmf": binom.pmf(k_range, n_trials, p_success), "poisson_pmf": poisson.pmf(k_range, lam), "lambda": lam, }
result = compare_pmfs(n_trials=100, p_success=0.05)
print(f"Binomial(100, 0.05) vs Poisson(lambda={result['lambda']}):")
print(f"{'k':>4} {'Binom PMF':>12} {'Poisson PMF':>13} {'Abs diff':>10}")
for k, b_pmf, p_pmf in zip(result["k"][:10], result["binomial_pmf"][:10], result["poisson_pmf"][:10]):
    print(f"{k:>4} {b_pmf:>12.6f} {p_pmf:>13.6f} {abs(b_pmf - p_pmf):>10.6f}")

Source: AIO p. 85

#probability-statistics#statistical-operations-and-random-variables#discrete-random-variables-drvs