- 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?

- 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?

- 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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