1. The problem statement, all variables and given/known data
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).
2. Relevant equations
f(y)=1/Θ and F(y)=y/Θ
3. 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.
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).
2. Relevant equations
f(y)=1/Θ and F(y)=y/Θ
3. 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.
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