NEED HELP with Rayleigh distribution evercise!!

arimya

New Member
Hi all!! I have some doubts in this exercise.

A RV X, has a rayleigh distribution if pdf takes the form fx(x) = x/theta (exp(-x^2/2theta)) where x>0 and theta is positive parameter. We know that E(X) = (theta pie/2) ^1/2, E(X^2) = 2 theta.
If the method of moments estimator is such that theta = (2/pie)(Xbar^2) and maximum likelihood estimator is 1/2n (Sum xi)^2 ,
Q1. how can I find the bias of the method of moment estimator and also show that the maxmum likelihood estimator is unbiased for theta
Q2. how can I write the mle for E(X)

I have done the following: bias(thetaestimator)=E(thetaestimator)- theta= E((2/pie)Xbar^2) -theta=(2/pie)E[Xbar^2]-theta= (2/pie)2theta-theta= ((4/pie)-1)theta.
and for seeing if the mle is unbiased: we have to see that E[thetahat]=theta so: E[ 1/2n (Sum xi)^2]= (1/2n)E[(Sum xi)^2]=(1/2n)nE[Xbar^2]=(1/2n)n2theta=theta and therefore in unbiased.

Are these solutions correct?
How could I do the Question2??
Thanks