Please answer the following :

1. Two teams, A and B, play a soccer match. The number of goals scored by Team A is modelled by a Poisson process {NA(t),t Z 0} with rate AA = 0.02 goals per minute. The number of goals scored by Team B is modelled by a Poisson process {NB(t),t 2 0} with rate A3 = 0.03 goals per minute. The two processes are assumed to be independent of each other. Let N {t} be the total number of goals in the match up to and including time t (measured in minutes). The soccer match itself is 90 minutes in length. [2] (a) Calculate the probability of the nal score being Team A: 1 Team B: 2. [2] (b) If Team A does not score in the opening 5 minutes of the match, what is the probability that Team A scores its rst goal during the rst half of the match? [4] (c) Calculate the probability that at least one goal is scored in the match and the rst goal is scored by Team B. [3] (d) Calculate Var(N3(90)|NB(4U) = 3). [3] (e) Calculate E[N(90)|NA(EIO) = 2]. [5] (f) If a total of 2 goals were scored in the match1 what is the joint probability that the rst goal took place in the opening 10 minutes of the match and the second goal was scored prior to the last 15 minutes of the match