# Expectation of Negative Binomial Distribution

## Main Question or Discussion Point

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?

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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?

D H
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 $\frac r p$.