# Expectation of Negative Binomial Distribution

1. Apr 21, 2008

### 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?

2. Apr 21, 2008

### 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?

3. Apr 21, 2008

### D H

Staff Emeritus
Both are correct!

BTW, I think you are using a very non-standard definition. r is almost always the predetermined number of successes that must be reached, not the number of trials.

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 $\frac r p$.

4. Apr 22, 2008

### GregA

Excellent!...thanks for that..by the way I ought to have said successes but it was late when I typed it...again, thanks :)