Negative Binomial Distribution

🎯 Negative Binomial Distribution

The Negative Binomial distribution models the number of trials needed to achieve a fixed number of successes in independent Bernoulli trials.

📌 Definition

Let $X \sim \text{NegBin}(r, p)$, where:

• $r$ is the number of successes we are waiting for,
• $p$ is the probability of success on each trial.

Then:

\[P(X = k) = \binom{k - 1}{r - 1} p^r (1 - p)^{k - r}, \quad k = r, r + 1, \dots\]

Alternatively, some definitions let $X$ count failures before the $r$-th success:

\[P(X = k) = \binom{k + r - 1}{r - 1} p^r (1 - p)^k, \quad k = 0, 1, 2, \dots\]

📋 Key Properties

Property	Value
Parameters	$r \in \mathbb{N}$, $p \in (0, 1]$
Support	$k \in {r, r + 1, \dots}$ or ${0, 1, \dots}$ (alternate)
Mean	$\mathbb{E}[X] = \frac{r}{p}$
Variance	$\text{Var}(X) = \frac{r(1 - p)}{p^2}$
Mode	$\left\lfloor \frac{(r - 1)(1 - p)}{p} \right\rfloor + r$
MGF	$M_X(t) = \left(\frac{p e^t}{1 - (1 - p)e^t}\right)^r$ for $t < -\log(1 - p)$

🌍 Real-World Examples

1. Customer Signups
   How many trials (ads, emails, visits) are needed to get 5 customer signups, if each contact has a 10% chance of converting?

2. Manufacturing
   How many defective parts will be found before identifying 3 non-defective parts on a faulty production line?

Notes

1. Generalizes the geometric distribution: Geometric is a special case with $r = 1$.
2. Models waiting time for $r$ successes.
3. Often used when data shows overdispersion (variance > mean).
4. Can be parameterized in terms of failures or trials, depending on context.
5. Appears in count models in statistics, e.g., in negative binomial regression.

🐍 Python Example

from scipy.stats import nbinom

# r = number of successes, p = probability of success
r, p = 3, 0.4

# SciPy's nbinom counts failures before the r-th success
rv = nbinom(r, p)

# PMF at a specific value (e.g., 5 failures before 3 successes)
print("P(X=5):", rv.pmf(5))

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

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