Sampling Distribution of sample mean for Poisson Distribution

Question

I am particularly struggling with part b, I don't know where to begin. For part a, I think the answer is that the sampling distribution is a Poisson(n$\lambda$).

Answer 1

Your answer is related to this subject

https://en.wikipedia.org/wiki/List_of_convolutions_of_probability_distributions

Since each variable in the sample is Poisson($\lambda$) distributed(and they are essentially independent), the sum would be, as you said, Poisson($n \lambda$).

The proof is simply deduced from the fact that the characteristic function of sum of independent random variables is the multiplication of their characteristic function. And using the inverse formula gives its density function(in case of discrete random variables their mass probability function).

Well the section(b) is not much more different.

Answer 2

The sampling distribution of a Poisson(λ) distributed random variable is given by:

$$ P(X_i = x_i) = f(x_i) = \frac{e^{-\lambda}\lambda^{x_i}} {x_i!} $$ where $x_i \in \{0, 1, 2 ...\}$

As you have correctly suggested the sum is Poisson(nλ) and therefore, substituting nλ for λ and $\Sigma x_i$ for $x_i$ the sampling distribution for $\Sigma X_i$ is given by:

$$ P(\Sigma X_i = \Sigma x_i) = g(\Sigma x_i) = \frac{e^{-n\lambda}(n\lambda)^{\Sigma x_i}} {(\Sigma x_i)!} $$ where $\Sigma x_i \in \{0, 1, 2 ...\}$

the sampling distribution for the sample mean, $\bar{X}$, is derived using the relation $\Sigma X_i =n\bar{X} $ as follows:

$$ h(\bar{x}) = P(\bar{X} = \bar{x}) = P(\Sigma X_i = n\bar{x}) = g(n\bar{x}) $$

$$ = \frac{e^{-n\lambda}(n\lambda)^{n\bar{x}}} {(n\bar{x})!} $$ where $\bar{x} \in \{0, \frac{1}{n}, \frac{2}{n} ...\}$

For $ n \ne 1$ This is not a poisson distribution as the pdf is not of the form $ \frac{e^{-\lambda}\lambda^{\bar{x}}} {\bar{x}!}$.

The sum distribution is stretched by factor 1/n as can be seen in these plots

(Note that the probabilities are only defined for the plotted circles, trendlines are just for visualising).

N = 2, λ = 3

N = 10, λ = 2