# Expectation of Negative Binomial Distribution

GregA
I am re-writing up some lecture notes and one of the proofs that E[X] for the negative binomial is r/p where r is the number of trials...The problem is there are a number of books that say r(1-p)/p is the correct expectation whilst others agree with 1/p

Which one is correct...for what its worth I have worked through the proof that the expectation for the geometric distribution is 1/p and find it pretty convincing...hence I'm lead to believe that r/p is correct for the negative binomial...I am worried though that a good number of sources differ in opinion however.

Can you folks pitch in please?

exk
wikipedia says r(1-p)/p for E[X] for a Negative Binomial Distribution. I tend to trust it on math topics.

Are you sure you wrote down the proof correctly?

Staff Emeritus
The reason both are correct is because there are two different interpretations of the random variable X: The number of failures that occur in reaching the r successes, or the number of trials needed to reach r successes. The expected value of the number of failures is $r\frac{1-p}{p}$ while the expected value of the number of trials is $\frac r p$.