43 Let X and Y be independent random variables Determine the

43. Let X and Y be independent random variables. Determine the distribution of (X -Y)/(X + Y) if (a)X,Y Epsilon Exp(1),

Solution

Let (lower-case) w be a number between 0 and 1

X – Y /X+Y < w if and only if (1w)X(1+w)Y.

We will find this probability.

The conditional probability that Y(1w)X/(1+w), given the value of X, is

e^(1w)X/(1+w).

The probability we seek is then the expected value of that:

E[e^(1w)X/(1+w) ] =e^(1w)x/(1+w) e^xdx =e^2x/(1+w)dx= (1+w)/2.

In other words, this random variable is uniformly distributed between w and 1.