h5soln - Homework 5 Solutions The George Washington...

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Homework # 5 Solutions The George Washington University ApSc 116 1 / 11
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Problem (Page 157 # 1, Section 3.7) Suppose that three random variables X 1 , X 2 , and X 3 have a continuous joint distribution with the following joint p.d.f.: f ( x 1 , x 2 , x 3 ) = ( c ( x 1 + 2 x 2 + 3 x 3 ) for 0 x i 1 i = 1 , 2 , 3 , 0 otherwise. Determine (a) the value of the constant c; (b) the marginal joint p.d.f. of X 1 and X 3 ; and (c) Pr ` X 3 < 1 2 ˛ ˛ X 1 = 1 4 , X 2 = 3 4 ´ . Solution: (a) We have Z 1 0 Z 1 0 Z 1 0 f ( x 1 , x 2 , x 3 ) dx 1 dx 2 dx 3 = 3 c . Since the value of this integral must be equal to 1, it follows that c = 1 / 3. (b) For 0 x 1 1 and 0 x 3 1, f 13 ( x 1 , x 3 ) = Z 1 0 f ( x 1 , x 2 , x 3 ) dx 2 = 1 3 ( x 1 + 1 + 3 x 3 ) . (c) The conditional p.d.f. of x 3 given that x 1 = 1 4 and x 2 = 3 4 is, for 0 x 3 1, g ( x 3 | x 1 = 1 4 , x 2 = 3 4 « = f ` 1 4 , 3 4 , x 3 ´ f 12 ` 1 4 , 3 4 ´ = 7 13 + 12 13 x 3 . Therefore, Pr X 3 < 1 2 | X 1 = 1 4 , X 2 = 3 4 « = 1 2 0 7 13 + 12 13 x 3 « dx 3 = 5 13 . 2 / 11
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Problem (Page 157 # 2, Section 3.7) Suppose that three random variables X 1 , X 2 , and X 3 have a mixed joint distribution with p.f./p.d.f.: f ( x 1 , x 2 , x 3 ) = ( cx 1+ x 2 + x 3 1 (1 - x 1 ) 3 - x 2 - x 3 if 0 < x 1 < 1 and x 2 , x 3 ∈ { 0 , 1 } , 0 otherwise. (Notice that X 1 has a continuous distribution and X 2 and X 3 have discrete distributions.) Determine (a) the value of the constant c; (b) the marginal joint p.f. of X 2 and X 3 ; and (c) the conditional p.d.f. of X 1 given X 2 = 1 and X 3 = 1 .
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