Exercise 3: [12 marks]LetXandYbe two independent and identically distributed standard normalrandom variables. Let[0,2),UandVbe two random variables such that:U=XcosYsinV=Xsin+YcosShow that the joint probability density function of (U,V) isfU(u,v) =12exp[12(u2+v2)].Exercise 4: [20 marks]1. Alice has 2 jobs that she will process sequentially. The first job takes anexponential amount of time with a mean of 15 minutes. Once the first jobis completed, then we start working on the second job. The second jobalso takes an exponential amount of time to complete with a mean of 15minutes.

(a) LetTbe the total time to complete both jobs. Give the distributionofT, its mean and its standard deviation.

(b) Compute the probability that it would take more than 45 minutes tocomplete both jobs.

(c) Suppose that Bob takes an exponential amount of time to processboth jobs (i.e. both, not each) with a mean time of 30 minutes.Compute the probability that it would take Bob more than 45 min-utes to complete both jobs.

(d) Suppose that Alice and Bob start to work at the same moment. Givethe probability that Bob will complete both jobs before Alice.2.

Suppose that Alice that an exponential amount of time with a mean of 20minutes for the first job, but the second job takes an exponential amountof time with a minutes of 30 minutes. LetTbe the total time needed (inminutes) to complete both jobs.

(a) Give the moment generating function forT.

(b) ClearlyE[T] = 50, however use the moment generating function inpart (a) to show that this is indeed the mean ofT

.(c) What is the probability that the second job would take less time thanthe first job?