Expectation of $ X(X - 1) \ldots (X - k + 1) $, where $ X $ has a Poisson distribution.

Question

I am trying to calculate $$ \text{E}[X(X - 1) \ldots (X - k + 1)], $$ where $ \text{E} $ denotes the expectation operator and $ k \in \mathbb{N} $ is fixed.

I think I have to use the fact that the expectation of a sum of random variables is the sum of the expectations, but how can we apply it to this product?

Ron Gordon · Accepted Answer · 2013-03-25T20:06:05.370

10

For a Poisson distribution

$$E[g(X)] = \sum_{j=0}^{\infty} g(j) \frac{\lambda^j}{j!} e^{-\lambda}$$

Specifically, for the $g$ you specified, the sum begins at $j=k$ because $g(X) = 0$ when $X<k$:

$$\begin{align}E[g(X)] &= \sum_{j=k}^{\infty} \frac{j!}{(j-k)!} \frac{\lambda^j}{j!} e^{-\lambda}\\ &= \sum_{j=0}^{\infty} \frac{\lambda^{j+k}}{j!} e^{-\lambda}\\ &= \lambda^k\\ \end{align}$$

edited Mar 25 '13 at 20:06

answered Feb 01 '13 at 20:10

Ron Gordon

138,521

You might want to check your definition of $g$, it is not correct. – Feb 01 '13 at 20:21
The function $g$ is $j!/(j-k)!$, not $j!/k!$. – Feb 01 '13 at 20:30
@ByronSchmuland: Oy. Not one of my finest moments. Thanks again for keeping me honest. – Ron Gordon Feb 01 '13 at 20:34

score 4 · Answer 2 · 2013-02-01T20:46:36.570

4

Alternatively, the probability generating function $G$ for a Poisson random variable with mean $\lambda$ is $G(s)=E(s^X)=\exp(-\lambda(1-s))$. Differentiate $k$ times and set $s=1$ to get the answer.

edited Feb 01 '13 at 20:46

answered Feb 01 '13 at 20:12

Expectation of $ X(X - 1) \ldots (X - k + 1) $, where $ X $ has a Poisson distribution.

2 Answers2