Suppose $x_1$,$x_2$,$x_3$,…..,$x_n$ are i.i.d. random variables with a common density poisson(λ)
(I is an indicator function)
- Find an unbiased estimator for $λ^2$
E $[$$\left(\frac{2 }{e^{-λ}}\right)$ * I{$x_1$=2}] = E $[$$\left(\frac{2 }{e^{-λ}}\right)$] * E [ I{$x_1$=2}] = $\left(\frac{2 }{e^{-λ}}\right)$ * $\left(\frac{e^{-λ}* λ^2 }{2}\right)$ = $λ^2$
$\left(\frac{2 }{e^{-λ}}\right)$ * I{$x_1$=2} is an unbiased estimator for $λ^2$
Is this correct? My issue is in the first to equalities I may have done something wrong,
-I am self learning so I will appreciate if you good confirm.
Best Answer
No, we can't use that. After all, we don't know $\lambda$; we can't use it in the formula for an estimator. The only things we have to use as inputs for the formula are our observations $x_1,x_2,\dots,x_n$.
As a hint, if $x$ comes from a Poisson random variable with parameter $\lambda$, the expected value of $x$ is $\lambda$ and the expected value of $x^2$ is $\lambda^2+\lambda$. Work with the averages of the $x_i$ and of the $x_i^2$.