Negative Binomial, Hypergeometric, and Discrete Uniform

Negative Binomial Distribution

Definition

$X \sim \text{NegBin}(r, p)$ counts the number of Bernoulli trials until the $r$th success.

PMF

$P(X = n) = \binom{n-1}{r-1} (1-p)^{n-r} p^r, \quad n = r, r+1, r+2, \ldots$

Why this formula?

• Trial $n$ must be a success (the $r$th one)
• Exactly $r-1$ successes in the first $n-1$ trials: $\binom{n-1}{r-1}$ ways
• $n - r$ failures, $r$ successes

Range

$X \in \{r, r+1, r+2, \ldots\}$

Expectation and Variance

$E[X] = \frac{r}{p}$

$\text{Var}(X) = \frac{r(1-p)}{p^2}$

Relation to Geometric

$\text{Geo}(p) = \text{NegBin}(1, p)$

The geometric counts trials until the first success.

Example: Coin Flips

Flip a fair coin until the third heads. $X \sim \text{NegBin}(3, 0.5)$.

$E[X] = \frac{3}{0.5} = 6$

$P(X = 5) = \binom{4}{2} (0.5)^2 (0.5)^3 = 6 \times \frac{1}{32} = \frac{3}{16}$

Hypergeometric Distribution

Definition

$X \sim \text{Hyp}(N, n, m)$ counts the number of successes when drawing $n$ items without replacement from a population of $N$ containing exactly $m$ successes.

PMF

$P(X = i) = \frac{\binom{m}{i} \binom{N-m}{n-i}}{\binom{N}{n}}, \quad i = \max(0, n-N+m), \ldots, \min(m, n)$

Why this formula?

• Choose $i$ successes from $m$: $\binom{m}{i}$
• Choose $n-i$ failures from $N-m$: $\binom{N-m}{n-i}$
• Divide by total ways to choose $n$ from $N$: $\binom{N}{n}$

Range

X ranges from $\max(0, n - (N-m))$ to $\min(n, m)$.

Expectation

$E[X] = \frac{nm}{N}$

Variance

$\text{Var}(X) = \frac{nm}{N}\left(1 - \frac{m}{N}\right)\frac{N-n}{N-1}$

The factor $\frac{N-n}{N-1}$ is the finite population correction.

Example: Quality Control

A batch of 20 components contains 5 defective. Select 4 at random. What is the probability exactly 2 are defective?

$X \sim \text{Hyp}(20, 4, 5)$

$P(X = 2) = \frac{\binom{5}{2} \binom{15}{2}}{\binom{20}{4}} = \frac{10 \times 105}{4845} = \frac{1050}{4845} \approx 0.217$

Example: Card Hands

Deal 5 cards from a 52-card deck. Probability of exactly 2 hearts?

There are 13 hearts in the deck.

$P(X = 2) = \frac{\binom{13}{2} \binom{39}{3}}{\binom{52}{5}}$

Comparison with Binomial

Property	Binomial	Hypergeometric
Sampling	With replacement	Without replacement
Trials	Independent	Dependent
Usage	Large population	Finite population
Approximation	-	Binomial when $n/N$ small

When $N \gg n$: $\text{Hyp}(N, n, m) \approx \text{Bin}(n, m/N)$

Discrete Uniform Distribution

Definition

$X \sim \text{Unif}\{1, 2, \ldots, N\}$ takes each integer from 1 to N with equal probability.

PMF

$P(X = k) = \frac{1}{N}, \quad k \in \{1, 2, \ldots, N\}$

Parameters

Some versions use $\{a, a+1, \ldots, b\}$ with $N = b - a + 1$.

Expectation

$E[X] = \frac{N + 1}{2}$

Variance

$\text{Var}(X) = \frac{N^2 - 1}{12}$

Example: Die Roll

A fair die: $X \sim \text{Unif}\{1, 2, 3, 4, 5, 6\}$.

$E[X] = \frac{7}{2}, \quad \text{Var}(X) = \frac{35}{12}$

Example: Random Digit

A random digit from 0 to 9: $X \sim \text{Unif}\{0, 1, \ldots, 9\}$.

$E[X] = \frac{9}{2} = 4.5$

Example: Tank Problem

In the tank estimation problem, captured serial numbers are assumed to come from $\text{Unif}\{1, 2, \ldots, N\}$, where we want to estimate the unknown $N$.

Summary Table

Distribution	PMF	Mean	Variance
NegBin($r,p$)	$\binom{n-1}{r-1}(1-p)^{n-r}p^r$	$r/p$	$r(1-p)/p^2$
Hyp($N,n,m$)	$\frac{\binom{m}{i}\binom{N-m}{n-i}}{\binom{N}{n}}$	$nm/N$	$\frac{nm}{N}(1-\frac{m}{N})\frac{N-n}{N-1}$
Unif$\{1,\ldots,N\}$	$1/N$	$(N+1)/2$	$(N^2-1)/12$