Geometric distribution is a fundamental discrete probability distribution that models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. Understanding this distribution is crucial for statistics, quality control, and decision-makingin various fields where waiting time until success is important.

Geometric Distribution Formulas

The geometric distribution has three main formulas: P(X = k) = (1-p)^(k-1) × p for exact probability, P(X ≤ k) = 1 - (1-p)^k for cumulative probability, and P(X ≥ k) = (1-p)^(k-1) for at-least probability. The distribution is characterized by its mean μ = 1/p and variance σ² = (1-p)/p². The memoryless property means that the probability of success in future trials doesn't depend on past failures.

• Exact Probability: P(X = k) = (1-p)^(k-1) × p
• Cumulative Probability: P(X ≤ k) = 1 - (1-p)^k
• At Least Probability: P(X ≥ k) = (1-p)^(k-1)
• Mean: μ = 1/p
• Variance: σ² = (1-p)/p²

Sources & References