## Negative Binomial Distribution Calculator

This calculator is used to find the probability and cumulative probabilities for negative binomial random variable given the number of successes ($r$) and probability of success ($p$).

Negative Binomial Distribution Calculator
Number of successes (r):
Number of failures (x):
Probability of success (p):
Result
Probability : P(X = x)
Cumulative Probability : P(X ≤ x)
Cumulative Probability : P(X < x)
Cumulative Probability : P(X ≥ x)
Cumulative Probability : P(X > x)

## Definition of Negative Binomial Distribution

A discrete random variable $X$ is said to have Negative Binomial distribution with parameter $r$ and $p$ if its probability mass function is \begin{aligned} P(X=x)&= \binom{x+r-1}{r-1} p^{r} q^{x},\\ & \quad \quad x=0,1,2,\ldots; r=1,2,\ldots\\ & \quad\quad \qquad 0<p, q<1, p+q=1. \end{aligned}

where

• $r =$ number of successes,
• $x =$ number of failures before $r^{th}$ success,
• $p =$ probability of success,
• $q = 1- p =$ probability of failures.