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Unformatted text preview: Math331, Spring 2008 Instructor: David Anderson Section 10.2 Homework Answers
Homework: pgs. 424  425, #'s 4, 5, 7, 9. 4. Let X and Y denote the numbers of sheep and goats stolen, respectively. We want Cov(X, Y ) = E[XY ]  E[X]E[Y ]. I will first calculate E[X] and E[Y ]. To see these, note that X and Y are both hypergeometric random variables. Therefore, calculating E[X] directly and E[Y ] using the formula for the hypergeometric RV (just to show the two ways to do this) gives
4 E[X] =
x=0 x 7 1 7 x 13 20 / 4x 4 7 20 13 +3 / 3 4 2 7 20 13 +4 / 4 4 1 20 13 / 4 0 = = 7 20 13 +2 / 2 4 3 7 = 1.4. 5 48 8 E[Y ] = = = 1.6. 20 5 Now we need E[XY ]. This is E[XY ] =
x+y4 xy 7 x 8 y 5 20 / . 4xy 4 The only nonzero terms of the above are (x, y) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)}. Therefore, E[XY ] =
x+y4 xy 20 4 7 2 7 x 8 y 11 8 1 5 20 / . 4xy 4 7 8 5 +12 1 1 2 5 7 8 +22 1 2 2 7 8 5 +13 1 2 1 5 7 8 +31 0 3 1 7 1 5 0 8 3 5 0 = 1/ +2 1 = 168 95 = 1.76842. Therefore, Cov(X, Y ) = E[XY ]  E[X]E[Y ] = 1.76842  1.4 1.6 = .47157. 5. Note that Y = n  X. Therefore, E[XY ] = E[X(n  X)] = nE[X]  E[X 2 ]. 1 However, X is a binomial(n, p) RV and so E[X] = np and E[X 2 ] = n2 p2  np2 + np. Y is binomial(n, 1  p) and so E[Y ] = n(1  p). Therefore, E[XY ] = nnp  n2 p2 + np2  np = n2 p  n2 p2 + np2  np and Cov(X, Y ) = E[XY ]  E[X]E[Y ] = n2 p  n2 p2 + np2  np  npn(1  p) = n2 p  n2 p2 + np2  np  n2 p + n2 p2 = np2  np = np(1  p). 7. We have that X, Y are independent. Therefore, Cov(X, Y + Z) = E [(X  E[X])(Y + Z  (E[Y ] + E[Z]))] = E [(X  E[X])((Y  E[Y ]) + (Z  E[Z]))] = E [(X  E[X])(Y  E[Y ])] + E [(X  E[X])(Z  E[Z])] = E [(X  E[X])] E [(Y  E[Y ])] + Cov(X, Z) = Cov(X, Z), where the fourth equality follows by the independence of X and Y . 9. Simply calculating using definitions gives V ar(X  Y ) = E[((X  Y )  (E[X]  E[Y ]))2 ] = E[((X  E[X])  (Y  E[Y ]))2 ] = E[(X  E[X])2  2(X  E[X])(Y  E[Y ]) + (Y  E[Y ])2 ] = V ar(X) + V ar(Y )  2Cov(X, Y ). 2 ...
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This note was uploaded on 06/24/2008 for the course MATH 331 taught by Professor Anderson during the Spring '08 term at Wisconsin.
 Spring '08
 Anderson
 Math, Probability

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