Question

Children arrive at an ice-cream parlour as a non-homogeneous 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.

Each child is independently a girl with probability 2/3 and a boy otherwise. What is the probability that exactly 5 girls arrive between 5pm and 6pm? [2 marks] (e) Girls always buy a chocolate ice cream, but boys buy a chocolate ice cream only with probability 1/2. What is the probability that exactly 5 chocolate ice creams are sold between 1pm and 4pm? [2 marks] (f) If we know that precisely one child arrived in the first hour, what is the density function of its arrival time? [2 marks] (g) If we know that precisely two children arrived in the first hour, what is the density function of the arrival time of the taller one? (height and arrival time are uncorrelated) [2 marks] (h) If we know that precisely two children arrived in the first hour, what is the density function of the arrival time of the child who arrived first? [2 marks] (i) If we know that at least one child arrived in the first hour, what is the density function of the arrival time of the child who arrived first? [2 marks] (j) 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?

Answer 1