Question

Math/IEng 381 Homework Work on this with at most one partner Due: Monday, April 9 A friend of yours decides she would like to collect all 12 Funko Guardians of the Galaxy Mini Mysteryl action figures by making a sequence of purchases of the "blind" boxes where it is unknown which figu Galaxy statistics course, she asks you how many purchases she will likely need to make. She also wonders about the total cost- assume boxes cost $7.95 each. she likes the element of surprise). Each possible Guardians of the is equally-likely in any purchase. Knowing that you are taking a probability and cost art the boxes Goals (show your work!): 1. Compute the mean, u, and standard deviation, a, of the mumber of purchases to complete the collection of all 12. (You would then report to your friend that it will take, on average, about p purchases give or take around ? purchases.) 2. Compute the mean, ?, and standard deviation, ?, of the cost to complete the collection of all 12. (You would then report to your friend that it will cost, on average, about u give or take around o. Hint: The total number of boxes, T, required may be expressed as T-1+X, +X,X where 1 is the number of boxes you need to buy to get your first action figure, X, is the number of additional boxes (beyond the first) you need to buy to get your second action figure, X, is the number of additional boxes (beyond that to get the second figure) you need to buy to get your third action figure, . . ., X12 is the number of additional boxes (beyond that to get the eleventh figure) you need to buy to get your twelfh action figure. Now determine the probability mass functions for the independent random variables X,, X., X and compute the mean and variance. The standard deviation can be found, of course, by taking the square root of the variance.

Answer 1

Let the total number of boxes be T. Then T = 1 + X_2 + X_3 + ...+ X_12 Each X_i follows geometric distribution with theta = 1/12. Therefore E[X_i] = 1/theta = 12 var[X_i] = (1 - theta)/theta^2 = 132 mu_T = E[1 + X_2 + X_3 + ...+ X_12] = 1 + 12 + 12 + ... + 12 (11 times) = 133 var_T = var[1 + X_2 +X_3 + ... + X_12] = 0 + 11^2 times 132 Therefore sigma_T = squareroot 11^2 times 132 = 126.38 Let cost to complete be C. Then, C = 7.957 Therefore mu_c = 7.95 times mu_T = $ 1057.35 sigma_c = 7.95 times sigma_T = $ 1004.72