Employees of a large company all choose 1 of 3 levels of health insurance coverage, for which premiums, denote

Employees of a large company all choose 1 of 3 levels of health insurance coverage, for which premiums, denoted by x, are 1, 2, 3 respectively. Premiums are subject to a discount, denoted by y, of 0 for smokers and 1 for non-smokers. The joint probability density function of x and y is given by: p(x,y) = { [(x² +y²)/31] for x = 1, 2, 3, and for y = 0, 1 } 0 otherwise Calculate the Variance of (x – y), the total premium paid by a randomly chosen employee.
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First, find p(x,y): p(1,0) = 1/31; p(2,0) = 4/31; p(3,0) = 9/31; p(1,1) = 2/31; p(2,1) = 5/31; p(3,1) = 10/31. The problem does not give the statistical weights, such as how many people smoke and how many do not. Since Sum(p) = 1, every possible case is to be treated as equally weighted (1/6). The expected value is 1/6. So the variance is easily calculated to be: [Sum(over all different x and y) (p - 1/6)^2]/6 = [Sum(p^2 - p/3 + 1/36)]/6 = [(1+16+81+4+25+100)/31^2 - 1/3 + 1/6]/6 = (227/961 - 1/6)/6 = 401/34596
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