uniform distribution

[Sinistral]

[rohan]

hey from whr did u get this ques.???

[Amit Goyal]

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.

[Sinistral]

thanx a lot.

but just one small doubt. X & Y are independent fine. but how did u infer Y & T are also independent?
or it is inferred by looking at the marginal pdf f_y(y) (by integrating f(y,t) from 0 to ∞ which comes out to be 1)??

[Amit Goyal]

That's right. You find marginal pdf of Y from joint pdf of T and Y by integrating it wrt t and that will be 1. Thus, Y is uniform. For finding distribution of T do the same but with respect to y. It is clear that in this case joint is the product of the marginals. Hence, Y and T are independent.