The equation below indicates expected value of negative binomial distribution. I need a derivation for this formula. I have searched a lot but can't find any solution. Thanks for helping 🙂
$$
E(X)=\sum_{x=r}^\infty x\cdot \binom {x-1}{r-1} \cdot p^r \cdot (1-p)^{x-r} =\frac{r}{p}
$$
I have tried:
\begin{align}
E(X) & =\sum _{x=r} x\cdot \binom{x-1}{r-1} \cdot p^r \cdot (1-p)^{x-r} \\[8pt]
& = \sum_{x=r}^\infty x \cdot \frac{(x-1)!}{(r-1)! \cdot ((x-1-(r-1))!} \cdot p^r \cdot (1-p)^{x-r} \\[8pt]
& = \sum_{x=r}^\infty \frac{x!}{(r-1)!\cdot ((x-r)!} \cdot p^r \cdot (1-p)^{x-r} \\[8pt]
\Longrightarrow & \phantom{={}} \sum_{x=r}^\infty r\cdot \frac{x!}{r!\cdot (x-r)!}\cdot p^r \cdot (1-p{)}^{x-r} \\[8pt]
& = \frac{r}{p} \cdot \sum_{x=r}^\infty \frac{x!}{r!\cdot (x-r)!}\cdot p^{r+1}\cdot (1-p)^{x-r}
\end{align}
If the power of $p$ in the last equation were not $r + 1,$ I can implement Newton Binomial. So It will be true. But I am stuck here.
Best Answer
If you want to continue that derivation instead of using linearity of expectation on a sum of i.i.d. geometric random variables, then you can follow this; however, doing it this way is much more complicated than the method using the i.i.d. variables.
When you arrive at the step $\operatorname{E}(X) = \sum_{x\geq r} r \binom{x}{r} p^r (1 - p)^{x - r}$, we can use this fact about power series:
$$ \frac{1}{(1 - z)^{r + 1}} = \sum_{n\geq r} \binom{n}{r}z^{n-r}, \quad \text{for }\lvert z\rvert < 1. $$
If this fact is unfamiliar to you, then you can derive it from the geometric series $\frac{1}{1 - z} = \sum_{n\geq 0} z^n$ by differentiating both sides $r$ times and dividing by $r!$. Of course, we are tacitly assuming that $p \neq 0$ in order to use this. Otherwise, the event that we want to occur $r$ times could not occur at all!
It follows that
$$\begin{align*} \operatorname{E}(X) &= r p^r\sum_{x\geq r} \binom{x}{r} (1 - p)^{x - r} \\ &= rp^r \cdot \frac{1}{\big(1 - (1 - p)\big)^{r + 1}} \\ &= rp^r \cdot \frac{1}{ p^{r + 1}} \\ &= \frac{r}{p} \end{align*}$$
We can do something similar for the variance using the formula
$$\begin{align*} \operatorname{Var} X &= \operatorname{E}\big(X^2\big) - \big(\operatorname{E}(X)\big)^2 \\ &= \operatorname{E}\big(X(X + 1)\big) - \operatorname{E}(X) - \big(\operatorname{E}(X)\big)^2. \end{align*}$$
This means that
$$\begin{align*} \operatorname{Var} X &= \sum_{x\geq r} x (x + 1)\binom{x - 1}{r - 1} p^r (1 - p)^{x - r} - \frac{r}{p} - \frac{r^2}{p^2} \\ &= \sum_{x\geq r} r (r + 1)\binom{x + 1}{r + 1} p^r (1 - p)^{x - r} - \frac{r p + r^2}{p^2} \\ &= r(r + 1)p^r \sum_{x\geq r+1} \binom{x}{r + 1} (1 - p)^{x - (r + 1)} -\frac{r p + r^2}{p^2} \\ &= r(r + 1)p^r \cdot \frac{1}{\big(1 - (1 - p)\big)^{r + 2}} -\frac{r p + r^2}{p^2} \\ &= \frac{r^2 + r}{p^2} - \frac{rp + r^2}{p^2} \\ &= \frac{r (1 - p)}{p^2}. \end{align*}$$