The (independent) visitors of a certain Web site may be divided into two groups: those who arrived on this site voluntarily (type I) and those who arrived there by chance or by error (type 11). Let N(t) be the total number of visitors in the interval [0,t]. We suppose that {N(t),t ≥ 0} is a Poisson process with rate λ = 10 per hour, and that 80% of the visitors are of type I (and 20% of type II).
(a) Calculate the mean and the variance of the number of visitors of type I, from a given time instant, before a second type II visitor accesses this site.
(b) Calculate the variance of the total time spent on this site by the visitors arrived in the interval [0,1] if the time (in minutes) X1 (respectively, X11) that a type I (resp., type II) visitor spends on the site in question is an exponential random variable with parameter 1/5 (resp., 2). Moreover, we assume that X1 and X11 are independent random variables.
(c) Suppose that, actually, {N(t), t ≥ 0} is a nonhomogeneous Poisson process whose intensity function is
and (t+24n) = λ(t), for n = 1,2,... . Given that exactly one visitor accessed this site between 6 a.m. and 8 a.m., what is the distribution function of the random variable S denoting the arrival time of this visitor?