Problems involving two distributions

[Travis Cooper]

"Pat arrives at the bus stop at some time U, which is uniformly distributed between time 0 and time 1, and waits for a bus. The first bus arrives at time T which is exponentially distributed with mean 1/μ. Assume that U and T are independent. What is the probability that Pat catches the first bus?"

This is not a homework problem but a practice problem in the text which no answer is supplied for. I have the following integral:

Pr {U < T} = *Integral from 0 to infinity* Pr {U < T | U = s}f_U(s) ds

Please tell me how I can solve these kinds of problems.

 

[Enuma_Elish]

"The" rule to remember is a function of random variable(s) is a random variable itself, and its distribution must be derived from the distribution(s) of the known random variable(s). I would have approached this problem by defining S = T - U, derive its distribution, then calculate Prob{S > 0}.

EnumaElish
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I would definitely have logged in as EnumaElish had PF administration awarded that account the privilege of posting replies, after I reset my e-mail address Tuesday, October 28, 2008.