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Geometric Distribution

Definition / Introduction

  • The Geometric distribution is a Discrete Probability Distribution that models the number of Bernoulli trials needed to achieve the first success.
  • It's closely related to the Binomial distribution, but instead of fixing the number of trials and counting successes, we fix the number of successes (at 1) and count the trials.
  • Examples: Number of coin flips until the first Head appears, number of items inspected until the first defective one is found, number of times you attempt a task until you succeed for the first time.

Key Concepts

1. Setup

  • We assume a sequence of independent Bernoulli trials, each with the same probability of success \(p\).
  • Trials continue until the first success occurs.

2. Two Versions of the Geometric Random Variable

There are two common definitions for the Geometric random variable \(X\): * Version 1 (Number of Trials): \(X\) is the total number of trials up to and including the first success. Possible values: \(k \in \{1, 2, 3, ...\}\). * Version 2 (Number of Failures): \(Y\) is the number of failures before the first success. Possible values: \(k \in \{0, 1, 2, ...\}\). Note that \(Y = X - 1\).

We will primarily use Version 1 (Number of Trials) in this note, as it's slightly more common, but be aware of both when consulting other sources.

3. Probability Mass Function (PMF) - Version 1 (Trials)

  • Let \(X\) be the number of trials until the first success. For \(X\) to equal \(k\) (where \(k \ge 1\)), we must have \(k-1\) failures followed by one success.
  • The PMF is: $$ P(X=k) = (1-p)^{k-1} p \quad \text{for } k = 1, 2, 3, ... $$
  • Where:
    • \(k\) is the number of trials (\(k \ge 1\)).
    • \(p\) is the probability of success on a single trial (\(0 < p \le 1\)).
    • \((1-p)\) is the probability of failure.

(PMF for Version 2 (Failures \(Y=k\)): \(P(Y=k) = (1-p)^k p\) for \(k=0, 1, 2, ...\))

4. Parameter

  • The Geometric distribution has one parameter:
    • \(p\): The probability of success on each trial (\(0 < p \le 1\)).

5. Expected Value (Mean) - Version 1 (Trials)

  • The expected number of trials needed to get the first success is: $$ E[X] = \frac{1}{p} $$
  • Intuition: If the probability of success is \(p=0.1\) (10%), you expect to need \(1/0.1 = 10\) trials on average to get the first success.

(Expected Value for Version 2 (Failures): \(E[Y] = \frac{1-p}{p}\))

6. Variance - Version 1 (Trials)

  • The variance of the number of trials is: $$ Var(X) = \frac{1-p}{p^2} $$

(Variance for Version 2 (Failures): \(Var(Y) = \frac{1-p}{p^2}\) - Same variance)

7. Memorylessness Property (Discrete Analogue)

  • The Geometric distribution is the only discrete distribution with the memorylessness property.
  • Mathematically (for Version 1): \(P(X > k + j | X > k) = P(X > j)\) for integers \(j, k \ge 0\).
  • Intuition: If you've already performed \(k\) trials without success, the probability distribution for the additional number of trials needed to get the first success is exactly the same as the original distribution starting from scratch. The process "doesn't remember" the past failures.
  • Example: If flipping a coin until Heads, knowing you've already flipped 5 Tails doesn't change the probability distribution of how many more flips you'll need.

Connections to Other Topics

  • Related to the Bernoulli distribution (models each individual trial).
  • The [[09_Negative_Binomial_Distribution|Negative Binomial distribution]] generalizes the Geometric by modeling the number of trials needed for \(r\) successes (Geometric is Negative Binomial with \(r=1\)).
  • The Exponential distribution is the continuous analogue of the Geometric distribution, modeling waiting time instead of number of trials, and also possessing the memorylessness property.

Summary

  • Models the number of trials (\(k\)) needed for the first success in independent Bernoulli trials.
  • Parameter: \(p\) (success probability).
  • PMF (Version 1): \(P(X=k) = (1-p)^{k-1} p\) for \(k \ge 1\).
  • Mean (Version 1): \(E[X] = 1/p\).
  • Variance (Version 1): \(Var(X) = (1-p)/p^2\).
  • Key Property: Memorylessness (discrete version).
  • Related to Negative Binomial (r=1) and Exponential (continuous analogue).

Sources