Expectation of $\bar{Y}^3$

Question

How do I calculate $\mathbb{E}\left[\bar{Y}^3\right]$?

$Y$ has a Poisson distribution with mean $\lambda$ , and there are $n$ independent $Y$'s.

Do I find $\mathbb{E}\left[Y^3\right]$ first? How do I go from there then?

Answer 1

Well, it depends on whether by $\overline Y^3$ you mean $$A = \frac{1}{n} \sum_{i=1}^n Y_i^3,$$ or $$B = \left(\frac{1}{n}\sum_{i=1}^n Y_i\right)^3.$$ They are, of course, not the same; the first is the mean of cubes, and second is the cube of the mean.

In the first case, the linearity of expectation gives us $$\mathbb{E}[A] = \mathbb{E}[Y^3],$$ which is easy to compute for $Y \sim \operatorname{Poisson}(\lambda)$. In the second case, you need to observe that $$\sum_{i=1}^n Y_i \sim \operatorname{Poisson}(n\lambda),$$ i.e., the sum of $n$ IID Poisson random variables is itself Poisson with rate parameter $n\lambda$. Consequently, $$\mathbb{E}[B] = \frac{1}{n^3} \mathbb{E}\left[\left(\sum_{i=1}^n Y_i\right)^3\right].$$ This of course is just $$\frac{1}{n^3} \mathbb{E}[W^3],$$ where $$W \sim \operatorname{Poisson}(n\lambda).$$