An engineer has observed that the gap times between vehicles passing a certain point on a highway have an exponential distribution with mean 10 seconds. Find the a. probability that the next gap observed will be no longer than one minute. b. probability density function for the sum of the next four gap times to be observed. What assumptions are necessary for this answer to be correct? Solution a) PDF(x)=1/10*e^(-1/10x) CDF(x)=1-e^(-1/10x) P(X<1)= CDF(1)=1-e^(-1/10) b) the sum of the next four gaps has Erlang(4,1/10) distribution We have to assume that the gaps are independent distributed See http://en.wikipedia.org/wiki/Erlang_distribution.

1. An engineer has observed that the gap times between vehicles passing a certain point on a
highway have an exponential distribution with mean 10 seconds. Find the
a. probability that the next gap observed will be no longer than one minute.
b. probability density function for the sum of the next four gap times to be observed. What
assumptions are necessary for this answer to be correct?
Solution
a) PDF(x)=1/10*e^(-1/10x) CDF(x)=1-e^(-1/10x) P(X<1)= CDF(1)=1-e^(-1/10) b)
the sum of the next four gaps has Erlang(4,1/10) distribution We have to assume that the gaps are
independent distributed See http://en.wikipedia.org/wiki/Erlang_distribution