问题

The negative binomial rv and distribution are based on an experiment satisfying the following conditions:

1. The experiment consists of a sequence of independent trials.
2. Each trial can result in either a success ($S$) or a failure ($F$).
3. The probability of success is constant from trial to trial, so $P(S\ \text{on trial}\ i)=p$ for $i=1,2,3,\ldots$
4. The experiment continues (trials are performed) until a total of $r$ successes have been observed, where $r$ is a specified positive integer.

The random variable of interest is

$X$=the number of failures that precede the $r$th success; 

$X$ is called a negative binomial random variable because, in contrast to the binomial rv, the number of success is fixed and the number of trials is random.

Possible values of $X$ are $0,1,2,\ldots$. Let $nb(x;r,p)$ denote the pmf of $X$. The event $\{X=x\}$ is equivalent to $\{r-1$ $S$'s in the first $(x+r-1)$ trials and an $S$ on the $(x+r)$th trial $\}$.

\[
\begin{split}
nb(x;r,p)&=P(X=x)\\
&=P(r-1\ S\text{'s}\ \text{on the first}\ x+r-1\ \text{trials})\cdot P(S)\\
&=\binom{x+r-1}{r-1}p^{r-1}(1-p)^x\cdot p\\
&=\binom{x+r-1}{r-1}p^{r}(1-p)^x
\end{split}
\]

Thus we have

Prop. The pmf of the negative binomial rv $X$ with parameters $r=$number of $S$'s and $p=P(S)$ is

\[
nb(x;r,p)=\binom{x+r-1}{r-1}p^{r}(1-p)^x,\quad x=0,1,2,\ldots
\]

In some sources, the negative binomial rv is taken to be the number of trials $X+r$ rather than the number of failures.

In the special case $r=1$, the pmf is

\[
nb(x;1,p)=(1-p)^x p,\quad x=0,1,2,\ldots\tag{*}
\]

Both $X$=number of $F$'s and $Y$=number of trials $(=1+X)$ are referred to in the literatures as geometric random variables(几何随机变量), and the pmf (*) is called the geometric distribution（几何分布）.

Prop. If $X$ is a negative binomial rv with pmf $nb(x;r,p)$, then

\[
E(X)=\frac{r(1-p)}{p},\quad V(X)=\frac{r(1-p)}{p^2}
\]

Finally, by expanding the binomial coefficient in front of $p^r(1-p)^x$ and doing some cancellation, it can be seen that $nb(x;r,p)$ is well defined even when $r$ is not an integer. This generalized negative binomial distribution has been found to fit observed data quite well in a wide variety of applications.

Remark:

负二项分布也称帕斯卡分布（巴斯卡分布）

References:

The above content is copied from the following:

Proposition in section 5 of Chapter 3, BOOK:

《Probability and Statistics For Engineering and The Sciences》(Fifth Edtion) P.131
Author: Jay L. Devore