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December 23rd, 2016, 07:58 AM   #1
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Waiting time for n buses (pdf)

In order to go from city A to city B you have to take n buses. For each of them you have to wait at the bus stop some time $\displaystyle \tau_{i}, i=1,2,...,n$. In each case, the probability that a bus arrives in a time interval $\displaystyle [t, t + dt)$ is equal to $\displaystyle rdt$, independently for each interval of time. Let T be the total time you have to wait for buses. What is the probability density function (pdf) of T?

Last edited by mike22; December 23rd, 2016 at 08:01 AM.
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December 23rd, 2016, 02:44 PM   #2
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The density function for each stop is $\displaystyle re^{-rt}$. To get the density function for the sum of the weight times, you can use the characteristic functions (Fourier transform of the density). Since the weight times all have the density, get the nth power of the characteristic function and then take the inverse transform to get the density function of the sum.
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