Asymptotic distribution of sample mean of the sum of two poisson distribution

Question

I'm trying to calculate the asymptotic distribution of the sample mean of the sum of two Poisson distributions.

Sample 1 is of size N1, and is from a Poisson distribution with expectation $\mu_1$. Sample 2 is of Size N2, and is from a Poisson distribution with expectation $\mu_2$.

Now, I was able to derive that when $N_1 \uparrow \infty$, and $N_2 \uparrow \infty$ at the same rate, the sample mean of the sum of two samples will be $\frac{(N_1 \cdot \mu_1 + N_2 \cdot \mu_2)}{N_1 + N_2}$.

From here, I'm having trouble deriving the standard deviation of the asymptotic distribution of the sample mean. How would I go about it?

Answer 1

I'm going to introduce some notation to hopefully clear things up.

Let $X_1,X_2,\dots,X_{N_1}$ be the samples from the first Poisson distribution, which are independent, and let $S_{N_1}$ be their sum. Let $Y_1,Y_2,\dots,Y_{N_2}$ be the samples from the second Poisson distribution, which are also independent (and independent of the $X_i$'s), and let $T_{N_2}$ be their sum. The $X_i$'s all have mean and variance $\mu_1$ and the $Y_i$'s all have mean and variance $\mu_2$.

The sample mean is then the random variable: $$\frac{X_1+\cdots+X_{N_1}+Y_1+\cdots+Y_{N_2}}{N_1+N_2} = \frac{S_{N_1}+T_{N_2}}{N_1+N_2}.$$ You have found the mean of the sample mean (not the sample mean, which is again a random variable) to be $$E\left[\frac{X_1+\cdots+X_{N_1}+Y_1+\cdots+Y_{N_2}}{N_1+N_2}\right]=\frac{E[S_{N_1}]+E[S_{N_2}]}{N_1+N_2}=\frac{N_1 \mu_1+N_2\mu_2}{N_1+N_2}.$$ To find the standard deviation of the sample mean, first find the variance. The two facts you need to use are:

1. $\text{Var}(aX)=a^2\text{Var}(X)$ for any constant $a$ and random variable $X$.
2. If $X$ and $Y$ are independent, then $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)$.

Thus, \begin{align*} \text{Var}\left(\frac{X_1+\cdots+X_{N_1}+Y_1+\cdots+Y_{N_2}}{N_1+N_2}\right) &=\left(\frac{1}{N_1+N_2}\right)^2\text{Var}(X_1+\cdots+X_{N_1}+Y_1+\cdots+Y_{N_2})\\ &=\cdots (\text{use fact 2 now}) \end{align*} I'll let you finish from here.