18. Let X and Y be independent exponential random variables having respective rates and . Let...

18. Let X and Y be independent exponential random variables having respective rates ë and ì. Let /, independent of X, Y, be such that with probability ^

and define Æ by

Æ =

with probability X, \i I = \

- y, if / = ï

ë + ì

ë

ë Ë- ì

(a) Show, by computing their moment generating functions, that X - Y and Æ have the same distribution.

(b) Using the lack of memory property of exponential random variables, give a simple explanation of the result of part (a).