Proving the expected value of a term of a negative binomial random variable

Question

The Question says:

The negative binomial random variable $X ∼ NB(r, p)$ has probability mass function $$P(X=n) = \binom{n-1}{r-1}p^r(1-p)^{n-r} \ \forall n\geq r$$ Prove that $$E[X^k] = \frac{r}{p}{E[(Y-1)^{k-1}]}$$ where $Y ∼ NB(r + 1, p)$ and hence calculate the expectation and the variance of X.

I'm unsure what $E[X^k]$ means, I'm not sure what I'm being asked to prove and I do not know how to calculate the expectation or variance of X

Answer 1

$$E[X^k] =\sum_{x\geq r} x^k \binom{x-1}{r-1}p^r(1-p)^{x-r}$$

$$=\sum_{x\geq r} x^{k-1} x \binom{x-1}{r-1}p^r(1-p)^{x-r}$$

$$=\sum_{x\geq r} x^{k-1} r \frac{x!}{r!(x-r)!}p^r(1-p)^{x-r}$$

$$=\frac{r}{p}\sum_{x\geq r} x^{k-1} \binom{x}{r} p^{r+1}(1-p)^{x-r}$$

$$=\frac{r}{p}\sum_{x\geq r} x^{k-1} \binom{x+1-1}{r+1-1} p^{r+1}(1-p)^{x-r}$$

$Y=X+1$

$$=\frac{r}{p}\sum_{y\geq r+1} (y-1)^{k-1} \binom{y-1}{r+1-1} p^{r+1}(1-p)^{y-(r+1)}= \frac{r}{p}{E[(Y-1)^{k-1}]}$$