Re: uniform distribution

Posted by Amit Goyal on May 04, 2013; 3:57am
URL: http://discussion-forum.276.s1.nabble.com/uniform-distribution-tp7580364p7580366.html

Hi Sinistral,

One way to do this problem is the following. We will find the distribution of Y and then find its variance.
First figure out the distribution of X=-ln(U_1) and Z=-ln(1-U_2). Check that both are exponential(1). Note that Y = X/(X+Z). Also Y only takes values between 0 and 1. Let T = X+Z. Use independence of X and Z (because U_1 and U_2 are independent) and find the joint density of Y and T. Check that the joint density is f(y, t) = te^{-t}. This gives us Y is uniform U(0, 1) and T is Gamma(2, 1). Thus Var(Y) = 1/12.

Let me know if you want the detailed working.

P.S. : There are many ways to do this problem. I presented the one which in my view is quick.