PLEASE DO NOT COPY FROM CHEG

The 2016 paper Modeling Travel Time for Reliability Analysis in a Freeway Network by N. Wu and J. Geistefeldt suggests that highway travel times can be modelled using a shifted gamma distribution. That is, the time for travelling a specific piece of highway is modelled as T = X + l where the random variable X follows a gamma distribution with some parameters and and l is some constant. As an example they consider the distribution with l = 20, E(T) = 25, and V(T) = 16.

(a) (5 points) Find the parameters and for this specific example.

b) (5 points) Derive the probability density function of T for this specific example.

(c) (5 points) Derive the moment generating function of T for this specific example.

(d) (5 points) Use the moment generating function to confirm the mean and variance of T.