I have random variable X with uniform distribution on interval [0, B], thus it has pdf f(x)=1/B when 0<=x<=B and 0 otherwise. Then its dumulative distribution function is x/B, 1 for x>B and 0 if X<0. For the given uniform distribution the E(x)= m= B/2 and Var(x)= s^2 = B^(2)/12.

If i.i.d is taken from this distribution x1, x2,....xn. and for the estimation of the expected value m=B/2 we have defined estimotors m1= x bar the sample average and m2= M/2, where M is the sample maximum the largest element of the sample.

MY PROBLEM:

To show that the sampling distribution of M has the following pdf.

f(m)=(n/B) *(m/B)^(n-1)

I assume the way to start is with the cumulative distribution function

F(m)= Pr(m<M) =.... {DO NOT KNOW HOW TO GO ON....}

Moreover i am not sure whether i got it right for the E (m2) and the Var(m2)

Is it the E(m) = B/4.

Any help is welcome!

Many thanks