Negative Binomial Distribution
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What is the Negative Binomial Distribution?
The negative binomial distribution models the number of failures before achieving a specified number of successes in a sequence of independent Bernoulli trials. Each trial has the same probability of success (p).
Probability Mass Function:
Where:
- is the number of successes needed
- is the number of failures before achieving successes
- is the probability of success on each trial
Properties
- Mean:
- Variance:
How to Calculate Negative Binomial Distribution in R
R
library(tidyverse)
r <- 3 # number of successes needed
p <- 0.4 # probability of success on each trial
# Calculate P(X = 10)
prob_exact <- dnbinom(7, size = r, prob = p)
print(str_glue("P(X = 10): {prob_exact}"))
# Calculate P(X <= 12)
prob_cumulative <- pnbinom(9, size = r, prob = p)
print(str_glue("P(X <= 12): {prob_cumulative}"))
# Calculate mean and variance
mean <- r/p
variance <- r * (1-p)/(p^2)
print(str_glue("Mean: {mean}"))
print(str_glue("Variance: {variance}"))
# Plot PMF
x <- 0:20
pmf <- dnbinom(x, size = r, prob = p)
pmf_df <- data.frame(x = x, pmf = pmf)
ggplot(pmf_df, aes(x = x, y = pmf)) +
geom_col() +
labs(
x = "Number of failures before r successes",
y = "Probability",
title = "Negative Binomial Distribution PMF"
) +
theme_minimal()How to Calculate Negative Binomial Distribution in Python
Python
import scipy.stats as stats
import numpy as np
import matplotlib.pyplot as plt
# Parameters
r = 3 # number of successes needed
p = 0.4 # probability of success on each trial
# Calculate P(X = 10)
prob_exact = stats.nbinom.pmf(7, r, p)
print(f"P(X = 10): {prob_exact:.4f}")
# Calculate P(X <= 12)
prob_cumulative = stats.nbinom.cdf(9, r, p)
print(f"P(X <= 12): {prob_cumulative:.4f}")
# Calculate mean and variance
mean = r/p
variance = r * (1-p)/(p**2)
print(f"Mean: {mean:.4f}")
print(f"Variance: {variance:.4f}")
# Plot PMF
x = np.arange(0, 21)
pmf = stats.nbinom.pmf(x, r, p)
plt.figure(figsize=(10, 6))
plt.vlines(x, 0, pmf, colors="b", lw=2)
plt.plot(x, pmf, "bo", ms=8)
plt.xlabel("Number of failures before r successes")
plt.ylabel("Probability")
plt.title("Negative Binomial Distribution PMF")
plt.grid(True)
plt.show()