Textbook Discrepancy for Negative Binomial Distribution

Question

In my textbook, for negative binomial distrubtion where $T_r$ denote the number of trials until the $r^\text{th}$ success in Bernoulli(p) trials.

$$E(T_r) =\frac{r}{p}$$

$$\operatorname{SD}(T_r)=\frac{\sqrt{rq}}{p} = \frac{\sqrt{r(1-p)}}{p}$$

But on Wikipedia, it says the negative binomial distribution has

Which doesn't equate to the above.

Answer 1

Right at the top of the Wikipedia article it says:

Different texts 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.

If you take the negative binomial distribution to be the distribution of the number of trials needed to get $r$ successes with probability $p$ of success on each trial, then the expected number of such trials is $r/p.$ The set of possible numbers of trials is then $\{r,r+1,r+2,\ldots\}.$

If you take it to be the number of successes before the $r$th failure with probability $p$ of success on each trial, then the expected number of such successes is $pr/(1-p).$ The set of possible numbers of such successes is then $\{0,1,2,3,\ldots\}.$

One could also define it to be the number of failures before then $r$th success, and then the expected value is $r(1-p)/p,$ and there are several other conventions.

One interesting thing about starting at $0$ rather than at $r$ is that in that case the distribution is actually infinitely divisible and one can allow non-integer values of $r.$