Negative Binomial Distribution

Binomial Distribution until a set number of successes occur

independent trials (with replacement)

Probability Distribution Function

A random variable $X$ has a negative binomial distribution $X \sim N B (k, p)$ if and only if it has the pdf

f (x; k, p) = (\binom{x - 1}{k - 1}) p^{k} (1 - p)^{x - k}

for $x \geq k$

$f (x) \geq 0$
$\sum_{x = k}^{\infty} f (x) = \sum_{x = k}^{\infty} (\binom{x - 1}{k - 1}) p^{k} (1 - p)^{x - k} = 1$

k = number of successes
$x =$ number of failures before the kth success
$k - 1$ as the last success is already set at last spot

E | X | = \frac{k (1 - p)}{p} / \frac{k}{p} ? ?

σ^{2} = \frac{k}{p} (\frac{1}{p} - 1)

f (x; k, p) = (\binom{x + k - 1}{x}) p^{k} (1 - p)^{x}