Pascal Distribution

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

Note

The binomial distribution models the number of successes in a fixed number of independent trials, while the Pascal distribution models the number of trials needed to achieve a fixed number of successes.

Mean

$$\mu_X = \frac{r}{p}$$

Note

Suppose $R$ is the number of successes before the experiment results in $k$ failures.

$$\mu_R = \frac{kp}{1-p}$$

Here $p$ is the probability of a success in a single trial. $$

Variance

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