Order statistics

Question

Consider the following function:

X~U[1,4] and Y~[1,4]

X and Y are independent. Let U be the maximum of X and Y. Give the density function of U.

a)1/9(u-1), 1<u<5

b)2/9(u-1), 1<u<5

c) none of the given

d)2/27(u-1)^2 , 1<u<5



e) 1/27(u-1)^2, 1<u<5

Answer 1

Hi Brandile!

To determine the density of any order statistic, recall that the distribution for the k^th order statistic is:

f_X(k)(x) = n!/(k-1)!(n-k!) * [F(x)]^k-1[1-F(x)]^n-kf(x)

where f(.) is the density of X_(k).

In the case of a Uniform(1,4) random variable, the density is just 1/3 between 1 and 4 and 0 elsewhere, and the CDF is 0 for x<1, (x-1)/3 for 1≤x≤4, and 1 for x>4.

Plugging things in, you get that following density for the maximum between 1 and 4:

f_X(2)(x) = 2!/(2-1)!(2-2)! * [(x-1)/3]^2-1*[1-(x-1)/3]^2-2(1/3)

We can rewrite this to make it match the answers a little more obviously:

f_X(2)(x) = 2/9 * (x-1) for 1≤x≤4, and 0 elsewhere.