## 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 | |
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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.