$r=$		$p=$
$x=$		P(X = x)  = P(X ≤ x)  = P(X ≥ x)  =

This applet computes probabilities for the negative binomial distribution $$X \sim NB(r, p)$$ where $$X = \mathrm{the\ number\ of\ failures\ before\ the}\ r^{th}\ \mathrm{success}$$

Directions

• Enter the number of successes in the $r$ box.
• Enter the probability of success in the $p$ box.
• Hitting "Tab" or "Enter" on your keyboard will plot the probability mass function (pmf).

To compute a probability, select $P(X=x)$ from the drop-down box, enter a numeric $x$ value, and press "Enter" on your keyboard. The probability $P(X=x)$ will appear in the pink box. Select $P(X \leq x)$ from the drop-down box for a left-tail probability (this is the cdf).

Details

• $f(x)=P(X=x)={x+r-1 \choose r-1} (1-p)^x p^r$
• $x=0,1,2,\ldots$
• $0 < p \le 1$
• $\mu=E(X)=\frac{r(1-p)}{p}$
• $\sigma^2=Var(X)=\frac{r(1-p)}{p^2}$
• $\sigma=SD(X)=\sqrt{\frac{r(1-p)}{p^2}}$