1. ## Poisson Sampling

Hi, here's the question:
Let $\displaystyle X_1,X_2,...,X_n$ be independent r.v.s with the Poisson distribution of parameter $\displaystyle m > 0$.
Let $\displaystyle Y = (X_1 + ... + X_n)/n$. Find the support and probability mass function of the distribution of $\displaystyle Y$. Calculate $\displaystyle E[Y-m]$ and $\displaystyle E[(Y-m)^2]$.
Any help would be appreciated. Thanks.

2. I get $\displaystyle E [Y] = E [1/n(X_1 + ... + X_n)] = (1/n) E [(X_1 + ... + X_n)]= (1/n) nm = m$ using linearity. Also, $\displaystyle Var[Y] = m/n$. But this is no longer a Poisson since the mean and variance are different?? If that's true then $\displaystyle E[Y-m]=0$. Is this right?

3. Originally Posted by davidmccormick
I get $\displaystyle E [Y] = E [1/n(X_1 + ... + X_n)] = (1/n) E [(X_1 + ... + X_n)]= (1/n) nm = m$ using linearity. Also, $\displaystyle Var[Y] = m/n$. But this is no longer a Poisson since the mean and variance are different?? If that's true then $\displaystyle E[Y-m]=0$. Is this right?
Indeed $\displaystyle Y$ is no longer a Poisson r.v., but you could have seen that directly from the question about the support of $\displaystyle Y$. While $\displaystyle X_1,\ldots,X_n$ are integers, their mean $\displaystyle Y$ may not be an integer. The support of the distribution of $\displaystyle Y$ is "the set of possible values" for Y (You can remove the " " in the present situation). Therefore, it is...

And in order to determine the probability mass function of Y, you should first find that (or recall that) of $\displaystyle X_1+\cdots+X_n$. It is Poisson with parameter $\displaystyle nm$. Then giving the probability mass function of Y should be simple. Tell us if you still have problems.

(and yes of course $\displaystyle E[Y-m]=0$)

4. Just let $\displaystyle S_n=X_1+\cdots+X_n$ and find it's distribution.

Then insert the n, since $\displaystyle Y_n=S_n/n$