Geometric Distribution

Core Concepts

The geometric distribution answers a natural question: how long must we wait for the first success in a sequence of independent Bernoulli trials? It is the discrete analogue of the exponential distribution and the only discrete distribution possessing the memoryless property.

PMF Derivation and Verification

CDF

Mean and Variance: Derivation

The Memoryless Property

Hazard Function

The geometric distribution has a constant hazard rate — the probability of success on any trial, given that we haven't succeeded yet, is always . This is another manifestation of memorylessness and distinguishes the geometric from distributions with increasing or decreasing hazard rates.

Relationship to Other Distributions

Worked Example: Quality Control

Python Implementation

import numpy as np
from scipy import stats

np.random.seed(42)

# Simulate geometric random variables
p = 0.3
n = 10000
samples = np.random.geometric(p, size=n)

# Verify mean and variance
print(f"Geometric(p={p})")
print(f"  Empirical mean:     {np.mean(samples):.4f}  (theoretical: {1/p:.4f})")
print(f"  Empirical variance: {np.var(samples, ddof=0):.4f}  (theoretical: {(1-p)/p**2:.4f})")

# Verify memoryless property
for s in [5, 10, 20]:
    given_gt_s = samples[samples > s]
    empirical = np.mean(given_gt_s > s + 5)  # P(X > s+5 | X > s) ≈ P(X > 5)
    theoretical = (1-p)**5
    print(f"  P(X > {s+5} | X > {s}): empirical={empirical:.4f}, theoretical P(X>5)={theoretical:.4f}")

Python Implementation: Hazard Rate Verification

import numpy as np

np.random.seed(42)

# Verify constant hazard rate for geometric distribution
p = 0.25
n = 50000
samples = np.random.geometric(p, size=n)

print(f"Geometric(p={p}) — Hazard Rate Verification")
print(f"{'k':>4} {'P(X=k | X>=k)':>14} {'p (theoretical)':>16}")
print("-" * 36)

for k in [1, 2, 3, 5, 10, 20]:
    given_ge_k = samples[samples >= k]
    if len(given_ge_k) > 0:
        hazard = np.mean(given_ge_k == k)
        print(f"{k:>4} {hazard:>14.4f} {p:>16.4f}")

# Show that geometric inter-arrival times in Bernoulli process are geometric
print(f"\nBernoulli process inter-arrival times:")
bernoulli = np.random.binomial(1, p, size=10000)
successes = np.where(bernoulli == 1)[0]
inter_arrival = np.diff(np.concatenate([[-1], successes]))
print(f"  Mean inter-arrival: {np.mean(inter_arrival):.4f} (theoretical: {1/p:.4f})")

Key Takeaways