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!