Geometric Distribution

🎲 Geometric Distribution

The Geometric distribution models the number of trials until the first success in a sequence of independent Bernoulli trials.

📌 Definition

Let $X \sim \text{Geometric}(p)$. Then:

\[P(X = k) = (1 - p)^{k - 1} p, \quad k = 1, 2, 3, \dots\]

📋 Key Properties

Property	Value
Parameter	$p \in (0, 1]$
Support	$k \in {1, 2, 3, \dots}$
Mean	$\mathbb{E}[X] = \frac{1}{p}$
Variance	$\text{Var}(X) = \frac{1 - p}{p^2}$
Mode	$1$
MGF	$M_X(t) = \frac{pe^t}{1 - (1 - p)e^t}$, for $t < -\log(1 - p)$

🌍 Real-World Examples

1. Sales Calls
   Number of calls until the first sale is made, where each has a success rate of 10%.

2. Dice Rolls
   Number of rolls until the first six appears.

Notes

1. Memoryless: $P(X > m + n \mid X > m) = P(X > n)$.
2. Models “wait time” for first success.
3. Special case of the negative binomial with $r = 1$.
4. Can be shifted to start from $0$ (alternative definition).

🐍 Python Example

from scipy.stats import geom

p = 0.3
rv = geom(p)

# PMF at a specific value
print("P(X=4):", rv.pmf(4))

# Sample 10 values
print("Samples:", rv.rvs(size=10))

# Mean and variance
print("Mean:", rv.mean())
print("Variance:", rv.var())