(1) Children arrive at an icecream parlour as a nonhomogeneous Poisson process with rate t, where t is the time measured in hours between noon and 6pm. Justify all your answers and write down the derivations in detail. (a) What is the mean number of children between 2pm and 5pm? [2 marks] (b) What is the probability that exactly 10 children arrive between 2pm and 5pm? [2 marks] (0) If we know that exactly 20 children arrive between noon and 6pm, what is the probability that exactly 10 arrive between 2pm and 5pm? [2 marks] ((1) If we know that precisely two children arrived in the rst hour, what is the density function of the arrival time of the child who arrived rst? [2 marks] (e) Assume that each child independently buys a geometric random number of ice creams with mean 2. What is the mean and variance of the number of ice cream sold before 2pm? [2 marks]