Find the Unbiased Estimator (Poisson)

Question

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.

Answer 1

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 $x_i^2$.

Answer 2

Hint:

We know $V(X_i)=\lambda=E(X_i)$

$V(X_i)=E(X_i^2)-(E(X_i))^2=E(X_i^2)-\lambda^2=\lambda$

$E(X_i^2)=\lambda+\lambda^2$

Now what should be my $\alpha$ so that I have $E(X_i^2 \pm \alpha ) = \lambda^2$