Negative Binomial Distribution

Statements

Let $X$ be a discrete random variable with the

Parameters

• $n$: number of independent Bernoulli trials.
• $p$: probability of success.

The random variable is said to follow a negative binomial distribution if the probability of getting exactly $k\in {\mathbb{N}}_{[1,n]}$ failures before $r$ success are occurs.

Probability Mass Function

$$\begin{array}{rrcl}{p}_X(\cdot ,n,p):& \mathbb{N}& \to & \mathbb{R}\\ & k& \to & \left(\begin{array}{c}k+r-1\\ k\end{array}\right)(1-p{)}^k{p}^r.\end{array}$$

Moments

• Expected value: $E[X]=r(1-p)p$.
• Variance: $\text{Var}(X)=r(1-p)/p^2$.

Uncertainty Estimation

Todo

Relationship with other Distributions

When $r=1$, a negative binomial distributed random variable is geometrically distributed.