what is Negative Binomial Distribution

Question

I read about Negative Binomial Distribution, but did not understood. Can you explain in simple word and using examples?

Answer 1

The binomiale negative distribution can be seen as a generalization of the geometric distribution.

Let $p\in (0,1)$ and let's say we have a random experiment with two outcomes: - success occur with probability $p$ - failure occur with probability $1-p$ For example your roll a die and the success is obtaining $6$.

Now you repeat this experiment several times, independently, until you get a success (you roll the die until you get $6$). And you count the number of failures before you get the success. This random variable follows a geometric distribution with parameter $p$, denoted by $\mathcal{G}(p)$. It takes its values in $\{0,1,2,\dots\}$ and we have $$P(X = k)=p(1-p)^k$$

That was the geometric distribution. Now let's say that you want to get $4$ times a success. You repeat the experiment and you count the number of failures before you get $4$ times the $6$. This random variable follows a negative binomial distribution with parameters $4$ and $p$. If you want $n$ successes it is a negative binomial with parameters $n$ and $p$, denoted by $NB(n,p)$.

As a particular case, we have $NB(1,p) = \mathcal{G}(p)$.

The probability mass function is given, for $k\in\{0,1,2,\dots\}$, by $$P(X=k) = {k+n-1 \choose k}p^n(1-p)^k$$

We can generalize further to the case where $n$ is not an integer using the Gamma function.