Re: Maths doubt

Posted by Amit Goyal on Apr 14, 2014; 9:19am
URL: http://discussion-forum.276.s1.nabble.com/Maths-doubt-tp7586601p7586861.html

Y = max{X(1), X(2), ... , X(n)} where X(i)s are i.i.d U[0, 1].
To find E(Y). Let us first find the CDF of Y.
for 0 < y < 1,
F(y)
= Pr(Y <= y)
= Pr(max{X(1), X(2), ... , X(n)} <= y)
= Pr(X(1) <= y, X(2) <= y, ... , X(n)} <= y)
= Pr(X(1) <= y)Pr(X(2) <= y) ... Pr(X(n)} <= y) [Because X(i)s are independent]
= y^n [Because X(i)s are U[0, 1]]

PDF of Y is
f(y) = dF(y)/dy = ny^{n-1}

E(Y) = ∫yf(y)dy = n/(n+1)