Question

1. Use the Moneyball data set which contains selected statistics for Major League Baseball teams from 1962–2012. Based on historical data, the probability that in a given year the NYM will make the playoffs is p = 7/47 = 0.149. Let X be the discrete random variable that gives the total number of Playoffs made by NYM in the last 20 years, i.e., from 1993 to 2012.                                                                                                                     (12 Points)

1. What is the probability that the total number of playoffs made by NYM during this 20-year period is exactly three?
2. What is the probability that the total number of playoffs made by NYM is at most 3?
3. What is the probability that the total number of playoffs made by NYM is at most 18?
4. What is the probability that the total number of playoffs made by NYM is at least 15?
5. What is the expected number of playoffs that NYM will make in this 20-year period?
6. Find the variance of the number of playoffs that NYM is expected to make in this 20–year period?

Answer 1

X is the discrete random variable that gives the total number of Playoffs made by NYM in the last 20 years

probability that in a given year the NYM will make the playoffs is p = 7/47 = 0.149

Therefore X~Bin(20,0.149)

a) probability that the total number of playoffs made by NYM during this 20-year period is exactly three

=P[X=3]

b)  probability that the total number of playoffs made by NYM is at most 3

=0.652579

c)  probability that the total number of playoffs made by NYM is at most 18

= 1

d) probability that the total number of playoffs made by NYM is at least 15

= 0(approx)

e) E[X] = 20*0.149 = 2.98 ~3

f) V(X) = 20*0.149*(1-0.149=2.536