Here's the question:
Let x1,x2,x3 be a random sample from a distribution with a PDF: f(x) = 2x, 0<x<1.
a) find the probability that the largest of x1,x2,x3 is less than the distribution median.
b) Find Cov(x(2),x(3))
Using quantiles, I found the distribution median to be = sqrt(1/2);
I found the max to be the pdf 6x^5 0<x<1;
In order to solve this, P(6x^5 < sqrt(1/2)) solved for x. This gave me an integral 2x with bounds from 0 to (((sqrt(.5))/6))^(1/5)) yielding 0.42514
Did I go through this correctly?
For the Covariance I am also not confident.
using the definition that cov(x,y) = E[(x - E(x))(Y - E(y))]
I found to be something like E[(12x^3-12x^5-24/35)(6x^5-6/7)] which I calculated to be around 78/245.
Thanks in advance for you help!