The Negative Binomial Distribution (Edexcel A-Level Further Mathematics): Flashcards

Number of failures before achieving $r$ successes in Bernoulli trials

$P(Y=x)=\binom{x+r-1}{r-1}p^r(1-p)^x$

Fixed number of successes

Probability of success in a single trial

Number of failures before achieving $r$ successes

$\mu = \frac{r(1-p)}{p}$

$\sigma^2 = \frac{r(1-p)}{p^2}$

$P(X \geq r)$ where $X \sim \text{Bin}(n, p)$

NegBin: failures before $r$ successes; Bin: successes in $n$ trials

$Y \sim \text{NegBin}(r, p)$