Understanding the Geometric Distribution
The geometric distribution is a discrete probability distribution that models the number of Bernoulli trials required to achieve the first success. It’s like flipping a coin or rolling a die repeatedly until a specific outcome occurs.
When to Use It
Use a geometric distribution when: - Each trial is independent. - There are only two outcomes: success or failure. - The probability of success, denoted by \(p\), remains constant. - You’re interested in either: - The trial number of the first success, or - The number of failures before the first success.
Two Types of Geometric Distributions
There are two closely related versions:
| Type | Interpretation | Support | Probability Mass Function |
|---|---|---|---|
| Type I (Shifted) | Number of trials until first success | \(\{1, 2, 3, \dots\}\) | \(P(X = k) = (1 - p)^{k-1} p\) |
| Type II | Number of failures before first success | \(\{0, 1, 2, \dots\}\) | \(P(Y = k) = (1 - p)^k p\) |
Note: The term “geometric distribution” can refer to either one, so always clarify which form you’re using.
Example
Suppose you’re rolling a fair six-sided die until you get your first “1”: - Success probability: \(p = \frac{1}{6}\) - The number of rolls follows a geometric distribution.
Key Properties
The distributions differ in their expected values and variances:
| Type | Mean | Variance |
|---|---|---|
| Trials until success | \(\frac{1}{p}\) | \(\frac{1 - p}{p^2}\) |
| Failures before success | \(\frac{1 - p}{p}\) | \(\frac{1 - p}{p^2}\) |
Recursive Cumulants (Advanced)
Let \(\mu = \frac{1 - p}{p}\) be the mean of the failures-before-success distribution. The cumulants \(\kappa_n\) satisfy the recursive relation:
\[ \kappa_{n+1} = \mu(\mu + 1) \frac{d\kappa_n}{d\mu} \]
This relates each cumulant to its predecessor using the mean.
Quick Summary
- Geometric distributions model the number of trials or failures before the first success.
- The two forms look similar but describe different things—always specify which you’re using.
- These distributions are memoryless: the probability of success remains unchanged no matter how many failures have occurred.