# Negative Binomial Distribution MGF

The Negative Binomial Distribution Wikipedia page lists its Moment-Generation Function as $$(\frac{1 - p}{1 - pe^t})^r$$.

However, the Moment-Generating Function page lists the MGF of the Negative Binomial Distribution as $$(\frac{pe^t}{1 - e^t + pe^t})^r$$.

Desmos reveals these not to be algebraically equivalent. Which one is correct, and why the disparity?

• The roles of $p$ and $q=1-p$ have been somehow flipped. See the text box on the Wikipedia NB page that discusses this inconsistancy. Commented Oct 7, 2020 at 15:25

It is enough to do the case in which $$r=1.$$
• Let $$X$$ be the number of independent trials before the first success, with probability $$p$$ of success on each trial. Then $$X\in\{0,1,2,3,\ldots\}$$ and $$\Pr(X=x) = (1-p)^x p.$$ Then we have \begin{align} M_X(t) & = \operatorname E(e^{tX}) = \sum_{x=0}^\infty e^{tx} (1-p)^x p \\[8pt] & = \frac p {1 - (1-p)e^t} = \frac{1-q}{1-qe^t}. \end{align}
• Now let $$X$$ be the number of independent trials needed to get one success, with probability $$p$$ of success on each trial. Then $$X\in\{1,2,3,\ldots\}$$ and $$\Pr(X=x) = (1-p)^{x-1} p.$$ Then we have \begin{align} M_X(t) & = \operatorname E(e^{tX}) = \sum_{x=1}^\infty e^{tx} (1-p)^{x-1} p \\[8pt] & = \frac{pe^t}{1 - (1-p)e^t} = \frac{(1-q)e^t}{1 - qe^t}. \end{align}
Thus the one on the MGF page is correct if you consider the "negative binomial distribution" supported on the set $$\{r,r+1,r+2,\ldots\},$$ and the one on the NB page is correct if you consider the NB distribution supported on the set $$\{0,1,2,3,\ldots\},$$ and the roles of $$p$$ and $$q$$ are interchanged.
Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at $$k = 0$$ or at $$k = r,$$ whether $$p$$ denotes the probability of a success or of a failure, and whether $$r$$ represents success or failure, so it is crucial to identify the specific parametrization used in any given text.