- Two teams
*A*and*B*play a best-of-three games series; the series ends as soon as one team has won twice. The series starts at*A*â€s home field, then goes to*B*â€s, and then returns to*A*â€s (if a third game is played.) Assume that the team playing at home has a probability of winning

*p*= 0.5 + Î´, where Î´ (a small positive number) is the â€œhome field advantage.â€ There are no draws. Find:- (a) The PMF of
*N*, the total number of games played. How does*E*[*N*] vary with Î´? - (b) The probability that
*A*wins the series, as a function of Î´. If Î´ = 0.05, how much of an advantage does team*A*get from the series starting at their home field?

- (a) The PMF of
- Let
*X*âˆ¼Geometric(0.8), and*Y*=*min*(*X*, 3).- (a) Determine the PMF of
*Y* - (b) Calculate
*Î¼**Y*and Ïƒ*Y*. Compare these with*Î¼**X*and Ïƒ*X*; how and why do they differ?

- (a) Determine the PMF of
- Buses arrive at the station randomly and independently, at a rate of 1 per five minutes; the number that arrive in
*t*minutes is thus a Poisson random variable*B*with Î» = 0.2*t*.- (a) Calculate the probability that exactly two buses come in a 10 minute interval.
- (b) How much time should you allow so that there is a 0.99 probability that at least one bus will arrive?
- (c) Suppose that, over a 10 minute interval, one bus arrives during the first
*x*minutes and a second bus arrives during the remaining time. Calculate the probability of this event as a function of*x*;*x*does not have to be a whole number. Compare your result with the answer to (a); how and why do the differ?

- A general linear transformation
*Y*=*aX*+*b*scales the PMF of*X*by a factor*a*and shifts it such that*Î¼**Y*=*aÎ¼**X*+*b*. Demonstrate this by calculating the PMF of*Y*when*X*is Bernoulli(1/2). Plot both*P**X*(*x*) and*P**Y*(*y*), including the means on your plots. - If
*X*is a random variable with mean*Î¼**X*and standard deviation Ïƒ*X*, find the expected value, variance and standard deviation of the random variable*Y*, where:*Y*=*X*âˆ’*Î¼**X*Ïƒ*X*

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