STA ASSIGNMENTS

Problem Set #1

1. You are given the following information regarding the joint distribution of X (the age of a person) and Y (the number of days they choose to spend at Saylorville Lake).

Values of Y 0 1 2 3

20 0.25 0.04 0.01 0.00 Values of X 40 0.15 0.12 0.08 0.05

60 0.25 0.04 0.01 0.00

a. What are the marginal distributions of X and Y ?

b. Compute E(X) and E(Y ).

c. Compute σ2X and σ 2 Y .

d. Compute σXY and Corr(X, Y ).

e. Are X and Y independent?

f. What are the conditional means E(Y |X = 20), E(Y |X = 40), and E(Y |X = 60)? g. A randomly selected person reports that they have spent 2 days at Saylorville Lake. What is the

probability that they are 40?

h. Finally, suppose that time spent at Saylorville Lake costs $100 plus $25 per day. That is, if Z denotes the total travel expenditure of an individual, then Z = 100 + 25 × Y . What is the mean expenditure of individuals visiting Saylorville Lake and the standard deviation of these expenditures?

2. Compute the following probabilities:

a. If Y ∼ N (2, 25), then what is P r(Y > 4)? b. If Y ∼ N (7, 49), then what is P r(Y < 0)? c. If Y ∼ N (5, 4), then what is P r(3 < Y ≤ 7)? d. If Y ∼ N (5, 16), then what is P r(3 < Y ≤ 11)?

3. Compute the following probabilities:

a. If Y ∼ χ211, then what is P r(Y > 19.68)? b. If Y ∼ χ23, then what is P r(Y > 11.34)? c. If Y ∼ F4,20, then what is P r(Y > 2.25)? d. If Y ∼ F3,7, then what is P r(Y > 8.45)?

4. Suppose that Yi iid∼ N (5, 9), i = 1, . . . , n. Compute P r(4.7 < Ȳ ≤ 5.3) using the central limit theorem for

a. n = 9?

b. n = 36?

c. n = 900

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