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 Aâ€s home field, then goes to Bâ€s, and then returns to Aâ€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 Î¼Y and ÏƒY . Compare these with Î¼X and ÏƒX ; 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» = 0.2t.
(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 Î¼Y = aÎ¼X + b. Demonstrate this by calculating the PMF of Y when X is Bernoulli(1/2). Plot both PX (x) and PY (y), including the means on your plots.
If X is a random variable with mean Î¼X and standard deviation ÏƒX , find the expected value, variance and standard deviation of the random variable Y , where: Y = X âˆ’ Î¼X ÏƒX
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