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Unformatted text preview: 42 Solutions Manual for Statistical Inference The complete definition of FXY is x 0 or y 0 02 x y/8 + y 2 x/4 0 < x < 2 and 0 < y < 1 2 x and 0 < y < 1 FXY (x, y) = y/2 + y 2 /2 . 2 x /8 + x/4 0 < x < 2 and 1 y 1 2 x and 1 y d. The function z = g(x) = 9/(x + 1)2 is monotone on 0 < x < 2, so use Theorem 2.1.5 to obtain fZ (z) = 9/(8z 2 ), 1 < z < 9. 1 1 7 4.5 a. P (X > Y ) = 0 y (x + y)dxdy = 20 . b. P (X 2 < Y < X) = 0 y 2xdxdy = 1 . 6 4.6 Let A = time that A arrives and B = time that B arrives. The random variables A and B are independent uniform(1, 2) variables. So their joint pdf is uniform on the square (1, 2) (1, 2). Let X = amount of time A waits for B. Then, FX (x) = P (X x) = 0 for x < 0, and FX (x) = P (X x) = 1 for 1 x. For x = 0, we have
2 a 1 y FX (0) = P (X 0) = P (X = 0) = P (B A) =
1 1 1dbda = 1 . 2 And for 0 < x < 1,
2x 2 FX (x) = P (X x) = 1  P (X > x) = 1  P (B  A > x) = 1 
1 a+x 1dbda = x2 1 +x . 2 2 4.7 We will measure time in minutes past 8 A.M. So X uniform(0, 30), Y uniform(40, 50) and the joint pdf is 1/300 on the rectangle (0, 30) (40, 50).
50 60y 0 P (arrive before 9 A.M.) = P (X + Y < 60) =
40 1 1 dxdy = . 300 2 4.9 P (a X b, c Y d) = P (X b, c Y d)  P (X a, c Y d) = P (X b, Y d)  P (X b, Y c)  P (X a, Y d) + P (X a, Y c) = F (b, d)  F (b, c)  F (a, d)  F (a, c) = FX (b)FY (d)  FX (b)FY (c)  FX (a)FY (d)  FX (a)FY (c) = P (X b) [P (Y d)  P (Y c)]  P (X a) [P (Y d)  P (Y c)] = P (X b)P (c Y d)  P (X a)P (c Y d) = P (a X b)P (c Y d). 4.10 a. The marginal distribution of X is P (X = 1) = P (X = 3) = 1 and P (X = 2) = 4 marginal distribution of Y is P (Y = 2) = P (Y = 3) = P (Y = 4) = 1 . But 3 1 1 P (X = 2, Y = 3) = 0 = ( )( ) = P (X = 2)P (Y = 3). 2 3 Therefore the random variables are not independent. b. The distribution that satisfies P (U = x, V = y) = P (U = x)P (V = y) where U X and V Y is
1 2. The ...
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 Spring '12
 Dr.Hackney
 Statistics

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