Let X1 X2 Xn be n independent continuous random variable

Let X1, X2, . . . , Xn be n independent continuous random variables, each having a

uniform distribution over (0, 1). Let
Y = max{X1,X2,...,Xn}

find the probability density function of Y .

Solution

Given that X1,X2,...,Xn are independent continuous random variables having uniform distribution over (0,1) and Y=max{X1,X2,...,Xn}.

The cumulative distribution function Fy(x) = P(Y<x) = P(X1<x,...Xn<x)

= P(X1<x)...P(Xn<x) (Since Xi\'s are independent)

= F1(x)...Fn(x)

= xⁿ in (0,1)

We know if F(x) is the CDF then the probability density function f(x) = F\'(x)

Therefore, the probability density function of Y = nx^n-1 in (0,1)

= 0 otherwise.