5. Maria has to take two buses to get home. The wait times of the buses...

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5. Maria has to take two buses to get home. The wait times of the buses are indepen- dent, and each is a Geometric (p) random variable expressed in minutes. (a) What's Maria's expected total wait time? (b) What's the probability she will wait more than n minutes? (c) What's the probability that she waited x minutes on the first bus, given that her total wait time was exactly n? 6. X and Y are independent Poisson (A) random variables. Recall that E [X] = E[Y] = A and var (X) = var (Y) = A. Let a and b be two arbitrary real numbers. Let Z aX + bY. = (a) What is E [Z]? (b) What is var (Z)? (c) Show that if a = b = 1, then Z is also a Poisson random variable, with a different parameter. 5. Maria has to take two buses to get home. The wait times of the buses are indepen- dent, and each is a Geometric (p) random variable expressed in minutes. (a) What's Maria's expected total wait time? (b) What's the probability she will wait more than n minutes? (c) What's the probability that she waited x minutes on the first bus, given that her total wait time was exactly n? 6. X and Y are independent Poisson (A) random variables. Recall that E [X] = E[Y] = A and var (X) = var (Y) = A. Let a and b be two arbitrary real numbers. Let Z aX + bY. = (a) What is E [Z]? (b) What is var (Z)? (c) Show that if a = b = 1, then Z is also a Poisson random variable, with a different parameter.