Repeat independent Bernoulli trials until a total of $r$ successes is obtained. The key difference from a binomial distribution is that instead of setting the number of trials $n$ in advance, you set the number of successes $r$. However, the negative binomial is better thought of as a series of [[geometric distributions]]. - The number of successes $r$ is fixed in advance - Trials are identical and result in success or failure (i.e., Bernoulli trials) - Trials are independent ## Examples - When tossing a coin, how many tosses before you obtain 5 heads? ## Notation $X \sim NB(r,p)$ ## Probability Mass Function $P(X=k) = \binom{k+r-1}{r-1} * p^r * (1-p)^k$ ## Expectation $ E(X) = \frac{r(1-p)}{p}$ ## Variance $V(X) = \frac{r(1-p)}{p^2}$