Maximum Order Statistic Question

[daneault23]

Homework Statement

Let Yi∼iid,uniform[0,θ]. Let U=max{Yi}. Derive the distribution of U and give the value of any associated parameters. Also calculate E(U) and Var(U).

Homework Equations

f(y)=1/Θ and F(y)=y/Θ

The Attempt at a Solution

Since we have a product of iid random variables, we can multiply the cdf's a total of n times, giving us F(yn)=[F(y)]^n=(y/Θ)^n, so f(u)=n(y/Θ)^n-1, meaning U~Be(n,1) with α=n and β=1.

I'm stuck on the E(u) part. This is what I have, ∫(from 0 to Θ)of u*n(y/Θ)^n-1 du. Please help.

 

[Ray Vickson]

Homework Statement

Let Yi∼iid,uniform[0,θ]. Let U=max{Yi}. Derive the distribution of U and give the value of any associated parameters. Also calculate E(U) and Var(U).

Homework Equations

f(y)=1/Θ and F(y)=y/Θ

The Attempt at a Solution

Since we have a product of iid random variables, we can multiply the cdf's a total of n times, giving us F(yn)=[F(y)]^n=(y/Θ)^n, so f(u)=n(y/Θ)^n-1, meaning U~Be(n,1) with α=n and β=1.

I'm stuck on the E(u) part. This is what I have, ∫(from 0 to Θ)of u*n(y/Θ)^n-1 du. Please help.

You cannot hope to get a correct answer if you are careless. From $F_n(y) = (y/\theta)^n$ we have the density $f_n(y) = dF_n(y)/dy = (n/\theta) (y/\theta)^{n-1},$ which is not what you wrote. I don't know why you write f(u) instead of f(y).

Anyway, the answer is given by a simple, calculus 101 integral. Just write it out and think about it.

 

[daneault23]

You cannot hope to get a correct answer if you are careless. From $F_n(y) = (y/\theta)^n$ we have the density $f_n(y) = dF_n(y)/dy = (n/\theta) (y/\theta)^{n-1},$ which is not what you wrote. I don't know why you write f(u) instead of f(y).

Anyway, the answer is given by a simple, calculus 101 integral. Just write it out and think about it.

Ray, so we would have ∫from 0 to Θ of (y*n/Θ)(y/Θ)^(n-1) then correct?

 

[daneault23]

If I'm doing things correctly here, I get E(U) = (nΘ)/n+1 and with the usual calculations, Var(U)=(nΘ^2)/(n+2)-((nΘ)/(n+1))^2