Pascal Distribution

1 min read Updated Fri Apr 24 2026 10:47:59 GMT+0000 (Coordinated Universal Time)

Aka. negative binomial distribution. Consists of independent bernoulli trials. The experiment is continued until $r$ successes are achieved. Observed variable is the number of total trials ( $X$ ). Denoted by $b^*(x;r,p)$ .

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

Here:

$p$ - probability of success in a single trial
$r$ - number of successes required

Mean

$\mu_X$ means the expected number of trials until $r$ successes.

\mu_X = \frac{r}{p}

Variance

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