Geometric Distribution

The geometric distribution is a probability distribution used to model the number of Bernoulli trials required to achieve the first success. It is commonly used in scenarios where events occur independently with a constant probability of success.

Probability Mass Function (PMF)

The probability of observing the first success on the k-th trial is given by:

P(X = k) = (1 - p)^{k - 1} p

• X: The number of trials until the first success.
• p: The probability of getting success in a single trial.
• (1 - p): The probability of getting failure in a single trial.

Properties

• Mean: 1/p
• Variance: (1 - p) / p^2
• Memoryless Property: The probability of success remains the same regardless of previous failures.

Example

The following example simulates a geometric distribution in Python:

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats

# Define probability of success
p = 0.3

# Generate geometric-distributed data
samples = np.random.geometric(p, size=1000)

# Plot histogram
plt.hist(samples, bins=range(1, max(samples)+1), density=True, alpha=0.75, color='blue', edgecolor='black')

# Overlay theoretical PMF
x = np.arange(1, max(samples) + 1)
plt.plot(x, stats.geom.pmf(x, p), 'ro-', label='Theoretical PMF')

plt.xlabel('Number of Trials')
plt.ylabel('Probability')
plt.title('Geometric Distribution (p=0.3)')
plt.legend()
plt.grid()
plt.show()

The output will be a histogram showing the distribution of trials needed to achieve the first success, with an overlaid theoretical probability mass function: