Solving Two Distributions Probability Question

[Travis Cooper]

Homework Statement

"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?"

Homework Equations

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The Attempt at a Solution

"Pr {U < T} = $$\int$$(from 0 to infinity) Pr {U < T | U = s}f_U(s) ds"

 

[lippyka]

"Pr {U < T | U = s} = Pr {T > s} = 1 - Pr {T \leq s} = 1 - F_T(s) = 1 - (1 - e^{-\mu s})""= e^{-\mu s}""fU(s) = 1 from 0 to 1""Pr {U < T} = \int_0^1 e^{-\mu s} ds = 1 - e^{-\mu} "Therefore, the probability that Pat catches the first bus is 1 - e^{-\mu}.