Question

Romeo and Juliet have a date at a given time, denote that random variable X and Y is the amount of time where Romeo and Juliet are late respectively. Assume X and Y are independent and exponentially distributed with different parameters λ and μ, respectively. Find the PDF of X – Y.

Answer 1

Given the random variables X and Y. You can think of X and Y as waiting times for two independent events (say U and V respectively) to happen. Suppose we wait until the first of these happens. If it is U, then (by the lack-of-memory property of the exponential distribution) the further waiting time until V happens still has the same exponential distribution as Y; if it is V the further waiting time until U happens still has the same exponential distribution as X. That says that the conditional distribution of X−Y given X>Y is the distribution of X, and the conditional distribution of X−Y given X<Y is the distribution of −Y Since P(X>Y)=λ/(μ+λ), that says the PDF for X−Y is given by