finding the distribution function for a function of r.v.s

[baileyjanet]

I have Y1, Y2, ..., Yn, a r.s. of size n from a uniformly distributed population with

f(y) = 1/theta, on 0 <=y<=theta
f(y) = 0, elsewhere.

so F(y) = def. integral from 0 to y of 1/theta dtheta = y/theta, on 0<=y<=theta.

I also have Y(n), the max of the Yis, and
U, a function of Y(n), with U = Y(n)/theta.

Using the method of distribution functions, I found the distribution function of U
F(u) = P(U<=u) = P(Y(n)/theta <=u) = P(Y(n) <= theta*u) = Fymax(theta*u)

I know that the distribution function of Y(n) is Fymax(y) = [F(y)]^n

and the density function is the derivative with respect to y of [F(y)]^n

here's where I'm having trouble...

d/dy[F(y)]^n = n[F(y)]^(n-1)*f(y) = n(y/theta)^(n-1)*(1/theta)

this is supposed to simplify to (y/theta)^n.

I'm not seeing it.

Could you please help?

Thanks if you can see what I'm doing!

Janet

 

[royhaas]

Your density function is correct. The distribution function is (y/theta)^n.

 

[baileyjanet]

Thank you so much for your reply. Yes, I see now that my terminology was at fault.