# Thread: Order Statistics of Beta Function

1. ## Order Statistics of Beta Function

Hey, I need help from you on another problem.

Let X1, ... , Xn ~(iid, independent and identical distribution) Beta(Alpha, 1)
(a) Prove that X(n), the maximum of random variable Xi, also has a Beta Distribution and identify its parameter.

(b) Based on (a), What is E(X(n))?

I'm very sorry that I don't know how to write equation as others.
Anyway, let me show your work.

Let X(n)= A.
Then, FA(a), F of A of a = [FX(a)]^n , F of X of a to the n.
And fA(a)=n(FX(a))^(n-1)*fX(a)
But I can't find FX(a).

THank you.
892king

2. Originally Posted by 892king
Hey, I need help from you on another problem.

Let X1, ... , Xn ~(iid, independent and identical distribution) Beta(Alpha, 1)
(a) Prove that X(n), the maximum of random variable Xi, also has a Beta Distribution and identify its parameter.

(b) Based on (a), What is E(X(n))?

I'm very sorry that I don't know how to write equation as others.
Anyway, let me show your work.

Let X(n)= A.
Then, FA(a), F of A of a = [FX(a)]^n , F of X of a to the n.
And fA(a)=n(FX(a))^(n-1)*fX(a)
But I can't find FX(a).

THank you.
892king
General result:

The pdf of $X(n) = \text{max} \, (X_1, \, X_2, \, ....... \, X_n)$ where $X_1, \, X_2, \, ....... \, X_n$ are i.i.d. is $f_n(x) = n [F(x)]^{n-1} f(x)$ where F(x) is the cdf of X and f(x) is the pdf of X.

For X ~ Beta $(\alpha, 1), ~ f(x) = \alpha x^{\alpha - 1}, ~ 0 \leq x \leq 1$ and zero elsewhere.

Therefore $F(x) = \alpha \int_0^x t^{\alpha - 1} \, dt = x^{\alpha}, ~ 0 \leq x \leq 1$.

Therefore $f_n(x) = n (x^{\alpha})^{n-1} \alpha x^{\alpha - 1} = n \alpha x^{n \alpha - 1}, ~ 0 \leq x \leq 1$ and zero elsewhere.

That is, $f_n(x) = \text{Beta} \, (n \alpha, 1)$.