Negative Binomial Distribution Calculator

Calculator

Parameters

Number of successes needed (r):

Probability of success (p):

Calculation Type:

Number of failures (X):

Distribution Chart

Enter parameters and calculate to view the distribution

Learn More

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:

P(X = k) = \binom{k+r-1}{k} p^r (1-p)^k

Where:

$r$ is the number of successes needed
$k$ is the number of failures before achieving $r$ successes
$p$ is the probability of success on each trial

Properties

Mean: $E(X) = \frac{r(1 - p)}{p}$
Variance: $Var(X) = \frac{r(1 - p)}{p ^ 2}$

Applications

1. Quality Control

Used to model the number of defective items encountered before finding a specified number of non-defective items in manufacturing.

2. Clinical Trials

Models the number of unsuccessful trials before achieving a target number of successful treatments in medical research.

3. Marketing

Analyzes the number of non-converting customers before achieving a specified number of sales.

4. Reliability Testing

Models the number of failures before achieving a set number of successful tests in product development.

R Implementation

1# Parameters
2r <- 3    # number of successes needed
3p <- 0.4  # probability of success on each trial
4
5# Calculate P(X = 10) - probability of needing exactly 10 trials
6prob_exact <- dnbinom(7, size = r, prob = p)
7print(paste("P(X = 10):", prob_exact))
8
9# Calculate P(X <= 12) - probability of needing 12 or fewer trials
10prob_cumulative <- pnbinom(9, size = r, prob = p)
11print(paste("P(X <= 12):", prob_cumulative))
12
13# Calculate mean and variance
14mean <- r/p
15variance <- r * (1-p)/(p^2)
16print(paste("Mean:", mean))
17print(paste("Variance:", variance))
18
19# Plot PMF
20x <- 0:20
21pmf <- dnbinom(x, size = r, prob = p)
22plot(x, pmf, type = "h", 
23     xlab = "Number of failures before r successes",
24     ylab = "Probability",
25     main = "Negative Binomial Distribution PMF")

Python Implementation

Python

1import scipy.stats as stats
2import numpy as np
3import matplotlib.pyplot as plt
4
5# Parameters
6r = 3    # number of successes needed
7p = 0.4  # probability of success on each trial
8
9# Calculate P(X = 10) - probability of needing exactly 10 trials
10prob_exact = stats.nbinom.pmf(7, r, p)
11print(f"P(X = 10): {prob_exact:.4f}")
12
13# Calculate P(X <= 12) - probability of needing 12 or fewer trials
14prob_cumulative = stats.nbinom.cdf(9, r, p)
15print(f"P(X <= 12): {prob_cumulative:.4f}")
16
17# Calculate mean and variance
18mean = r/p
19variance = r * (1-p)/(p**2)
20print(f"Mean: {mean:.4f}")
21print(f"Variance: {variance:.4f}")
22
23# Plot PMF
24x = np.arange(0, 21)
25pmf = stats.nbinom.pmf(x, r, p)
26plt.figure(figsize=(10, 6))
27plt.vlines(x, 0, pmf, colors="b", lw=2)
28plt.plot(x, pmf, "bo", ms=8)
29plt.xlabel("Number of failures before r successes")
30plt.ylabel("Probability")
31plt.title("Negative Binomial Distribution PMF")
32plt.grid(True)
33plt.show()