Question

The purchase of coffee from a cafe on campus follows a Poisson process with rate A 300 coffees per day. The price Y of each coffee purchased is uniformly distributed on (3.5, 4.5, 5.0. The total amount of money the cafe makes in time t is defined N(t) S(t)=Y i-1 YN (t) are where Yi, id according to Y. Recall EIS(t E[N(t)]E[Y] VARIS(t EIN(t)] VAR[Y] + VAR[N (t)] E [Y]2 What is the expected amount of money the cafe will make in one week? [2 marks] (a) What is the variance of the amount of money the cafe will make in one day? (b) [2 marks] (c) Would these results change for a Poisson process where A(t) depends on the time of day as well? 1 marks

Answer 1

NL) N(t)Poisson (=30 pur day) 3.5,5,5} e 7 time EINL4)7x3e0 = lov One week 35+455) 13 3 Var(NE() ENUD) .Var(4] + b) VanfsU Var(Y) EN) -E 125-169 =3.5 -51.5= 2. 5750 Ye te uutt woutd ihane a tts derivet ВНОGAL aing Tndcterdence