Question

the best selling basketball sneaker of all time is the converse 8820 chuck Taylor 8221 all star. suppose chuck Taylor 8221s are sold in holabird sports and are available in sizes from 7 to 13. the size purchased is a random variable with a probability distribution given in the table below.

x 7 8 9 10 11 12 13
p(x) 0.05 0.08 0.10 0.14 0.28 0.21 0.14

1. find the mean, variance and standard deviation of the sneaker size purchased

2. find the probability that a randomly selected customer buys a pair of these sneakers with size greater than one standard deviation to the right of the mean

3. suppose 3 randomly selected customers each purchase a pair of these sneakers. what is the probability that exactly 2 of the 3 buy size 11 sneakers

Answer 1

1) Mean = E(X) = 7 * 0.05 + 8 * 0.08 + 9 * 0.1 + 10 * 0.14 + 11 * 0.28 + 12 * 0.21 + 13 * 0.14 = 10.71

E(X²) = 7² * 0.05 + 8² * 0.08 + 9² * 0.1 + 10² * 0.14 + 11² * 0.28 + 12² * 0.21 + 13² * 0.14 = 117.45

Var(X) = E(X²) - (E(X))² = 117.45 - 10.71² = 2.7459

SD(X) = sqrt(2.7459) = 1.66

2) P(X > mean + sd) = P(X > 10.71 + 1.66) = P(X > 12.37) = P(X = 13) = 0.14

3) P(X = 11) = 0.28

n = 3

P(exactly 2 of 3 buy size 11 sneakers) = 3C2 * 0.28² * (1 - 0.28)^3-2 = 0.1693