1. ## proof consistency

A random variable X has a Poisson - distribution defined for each nonegative integer k, Pr(X= k) = (lamda^(k)/k!)*exp(-lamda). We know that E(X)= lamda and Var(X) = E(X^2) - E^(2) (X) = lamda..
We draw an i.i.d sample of size n from this distribution: x1, x2,..., xn, and define Zn = Sum (from i=1 to n) xi^(2). Show that lamda(kappa) = (-1+sqrt(1+4Zn)) /2 is a consistent parameter of Lamda.

I know that for consistency we have to estimate the expected value and the bvariane or find the plim estimator.
Unfortunatelly i cannot show it using either way. Please if you know show me the proof.

Many thanks!

2. Zn = Sum (from i=1 to n) xi^(2) goes to infinity almost surely.
So, something is wrong with this question.
You need to divide by n here or in this kappa thingy

${Zn\over n} = {\sum_{i=1}^n X_i^2\over n}\to \lambda +\lambda^2$ almost surely.

3. If such the case Zn= sum(of xi^2 for i=1 to infinity)/n
What do i show with that Zn is going to infinity. How is that relevant in showing that Lamda kappa is consistent estimator of parameter lamda.

Thank you.

4. Do i need to change inside lamda kappa that Zn is lamda^2 + lamda?

If i do it then i get that the expected value for lamda kappa is lamda. But i am stuck with the variance.
How much will be the variance for Zn? (not the one in the original problem but such as you have defiened it)

Thank you, again.