Mean and Variance of Negative Binomial Distribution

A negative binomial models the number of trials  
\[n\]
  to achieve a desired number  
\[x\]
  of successes.
We can treat the negative binomial as a sequence of geometric sequences each with parameter  
\[p\]
, each sequence ending in a success. The expected number of trial until a success is achieved in  
\[\frac{1}{p}\]
  and since we require  
\[x\]
  successes we would expect to have to carry out  
\[\frac{x}{p}\]
  trials.
Similarly the variance of a geometric distribution is  
\[\frac{1-p}{p^2}\]
  so the variance of a sequence of  
\[x\]
  such geometric distributions is  
\[\frac{x(1-p)}{p^2}\]
.

Add comment

Security code
Refresh