probability-theory-and-statistics-5

1. 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 + δ, where δ (a small positive number) is the “home field advantage.” There are no draws. Find:
   1. (a) The PMF of N, the total number of games played. How does E[N] vary with δ?
   2. (b) The probability that A wins the series, as a function of δ. If δ = 0.05, how much of an advantage does team A get from the series starting at their home field?
2. Let X ∼Geometric(0.8), and Y = min(X , 3).
   1. (a) Determine the PMF of Y
   2. (b) Calculate μY and σY . Compare these with μX and σX ; how and why do they differ?
3. 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 λ = 0.2t.
   1. (a) Calculate the probability that exactly two buses come in a 10 minute interval.
   2. (b) How much time should you allow so that there is a 0.99 probability that at least one bus will arrive?
   3. (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?
4. A general linear transformation Y = aX + b scales the PMF of X by a factor a and shifts it such that μY = aμX + 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.
5. If X is a random variable with mean μX and standard deviation σX , find the expected value, variance and standard deviation of the random variable Y , where: Y = X − μX σX

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