Verified answer

Let U = MAX{x1,x2,....xn}

P( U < k) = P( x1 < k) ∩ P(x2 < k) .....∩ P(xn < k) .= [F(k)]^n

F(k) =2 ∫ x dx , 0 < x < k = 2 k^2/2 = k^2

f(x) = n [ F(x)]^(n-1) F'(x)

f(x) = n f(x) [F(x)]^(n-1) ---------- distribution of the maximum of x = Xn

= n (2x) [x^2]^(n-1)

= 2 n x^(2n-1) , 0 < x < 1

Let Xn = U

The distribution of the maximum U is :

2n u^(2n-1) , 0 < u < 1

What is the probability that the biggest one is less than .8?

Integrate f(u) from 0 to 0.8

2n (.8)^(2n) /2n = (.8)^(2n)