Negative Binomial Distribution Reference Cheat Sheet

Key Facts

• If $X$ is the total number of trials needed to get $r$ successes with success probability $p$, then $X \sim \operatorname{NegBin}(r,p)$ has support $x=r,r+1,r+2,\ldots$.
• For the trials-counting form, the PMF is $P(X=x)=\binom{x-1}{r-1}p^r(1-p)^{x-r}$ for $x=r,r+1,r+2,\ldots$.
• If $Y$ is the number of failures before the $r$th success, then $Y=X-r$ and the support is $y=0,1,2,\ldots$.
• For the failures-counting form, the PMF is $P(Y=y)=\binom{y+r-1}{r-1}p^r(1-p)^y$ for $y=0,1,2,\ldots$.
• For $X$ equal to total trials, the mean is $E[X]=\frac{r}{p}$ and the variance is $\operatorname{Var}(X)=\frac{r(1-p)}{p^2}$.
• For $Y$ equal to failures before $r$ successes, the mean is $E[Y]=\frac{r(1-p)}{p}$ and the variance is $\operatorname{Var}(Y)=\frac{r(1-p)}{p^2}$.
• When $r=1$, the negative binomial distribution reduces to the geometric distribution, with $P(X=x)=(1-p)^{x-1}p$ for $x=1,2,3,\ldots$.
• The trials must be independent, each trial must have the same success probability $p$, and the process must stop when the $r$th success occurs.

Vocabulary

Bernoulli trial

A Bernoulli trial is a random trial with exactly two outcomes, usually called success and failure, where the success probability is $p$.

Success probability

The success probability $p$ is the fixed probability that one independent trial results in a success.

Negative binomial distribution

A negative binomial distribution gives probabilities for the number of trials or failures needed to obtain a fixed number $r$ of successes.

Support

The support is the set of possible values a random variable can take, such as $x=r,r+1,r+2,\ldots$ for total trials.

Probability mass function

A probability mass function assigns a probability $P(X=x)$ to each possible value of a discrete random variable.

Parameterization

A parameterization is the chosen way to define the random variable, such as counting total trials $X$ or failures $Y$ before the $r$th success.