1. Let the RV's X and Y have the joint pdf f(x,y): = 10 1/(1 - xy) 0 < x < 1,0 < y < 1 otherwise What is P(X ≥ 2Y)? Show by first integrating over x and then integrating over d cb y. That is, use an expression of the form få få f(x, y) dx dy where you fill in the constants a, b, c, d as appropriate. 2. Two construction companies make bids of X and Y on a remodeling project. The joint pdf of X and Y is uniform on the space 2 < x < 2.5,2 < y < 2.3. If X and Y are within 0.1 of each other, the companies will be asked to rebid; otherwise, the low bidder will be awarded the contract. (a) What is the joint pdf of X and Y? [Property b of a joint pdf will be helpful here.] (b) What is the probability that they will be asked to rebid? 3. Suppose, for the families of your country, Y is annual income (measured in thou- sands of $$) for a family and X is the number of children a family has. If a family's "income per child" (i.e., X) is less than $50,000 then they receive a $2,000-per-child tax credit. [For the purposes of this problem, treat both Y and X as continuous random variables, so that you may think of f(x, y) as a joint pdf.] (a) Do you think X and Y are correlated? Explain in two or so sentences. [You can't show this mathematically here! This question is asking you to think about the world. The important part of your response coming up with a reasonable explanation.] (b) Given your response above, do you think f(x,y) = ƒx(x)fy(y)? Explain. (c) Write an expression (using a double integral of course) for the share of families that qualify for the tax credit. [Your response will be in terms of f(x,y). You cannot provide a number here.] (d) Suppose there are N families. What is the expected cost to the government of the tax credit? 5. There are quite a few results about the distribution of a function of X = 1/1Σ₁ X; for i = 1,2,...,n. See if you can keep them straight: (a) When the distribution of each X; is N(μ,σ²), then what is a function of X that is exactly N(0,1)? What if any additional assumptions do you need? For ex- ample, do you require that n be large? Briefly explain. (b) When the distribution of each X; is NOT normal but does have a known mean E[X] = μ and known var(X;) = 0², then what is a function of X that is approx- imately N(0,1)? What additional assumptions do you need? For example, do you require that n is large? Briefly explain. (c) Now suppose the distribution of each X; is normal with mean μ but you don't know the variance. What is a function of X that is approximately N(0,1) that does NOT depend on the unknown o²? What additional assumptions do you need? For example, do you require that n be large? Briefly explain. (d) Consider the previous part where σ² is unknown but now suppose that n is small. What can you say about the distribution of (some function of) X? Briefly explain. 6. Let X1, X2, ..., X16 be a random sample from a normal distribution N(77,25). Com- pute (a) P(77 < X < 79.5) (b) P(74.2 < x < 78.4) 7. Let X equal the weight of the soap in a "6-pound" box. Assume X N(6.05, 0.0004). (a) Find P(X < 6.0171). (b) If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. Hint: Let Y equal the number of boxes that weigh less than 6.0171 pounds. (c) Let X be the sample mean of the nine boxes. Find P(X ≤ 6.035).