Pascal Distribution

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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}

Written by

Sahithyan K.

May 31, 2026 1 min read

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