An article in Mathematical Biosciences [“Influence of Delayed Viral Production on Viral Dynamics in HIV-1 Infected Patients” (1998, Vol.152(2), pp. 143–163)] considered the time delay between the initial infection by immunodeficiency virus type 1 (HIV-1) and the formation of productively infected cells. In the simulation model, the time delay is approximated by a gamma distribution with parameters r = 4 and 1/ λ = 0.25 days. Determine the following:

(a) Mean and variance of time delay

(b) Probability that a time delay is more than half a day

(c) Probability that a time delay is between one-half and one day