quizexe5 - (e What is the conditional pdf of X given Y =...

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STAT 427 Quiz /Class Exercise 5: Name: With Solution. Suppose that two continuous random variables X and Y have a joint probability density function (pdf) f ( x, y ) = ( 1 8 ( x + y ) , 0 < x < 2 , 0 < y < 2 0 , otherwise (a) What is P (0 < X < 1 , 0 < Y < 1)? (2 points) P (0 < X < 1 , 0 < Y < 1) = Z 1 y =0 ±Z 1 x =0 1 8 ( x + y ) dx ² dy = Z 1 y =0 " 1 8 ³ 1 2 x 2 + xy ´µ µ µ µ 1 x =0 # dy = Z 1 y =0 1 8 ( 1 2 + y ) dy = 1 8 ³ 1 2 y + y 2 2 ´µ µ µ µ 1 y =0 = 1 8 (b) Construct the marginal pdf of X , f X ( x ). (2 points) f X ( x ) = Z 2 y =0 1 8 ( x + y ) dy = 1 8 ( xy + y 2 2 ) µ µ µ µ 2 y =0 = x + 1 4 , 0 < x < 2 . (c) Compute the EX and V ar ( X ). (2 points) EX = Z 2 0 x x + 1 4 dx = 7 6 . EX 2 = Z 2 0 x 2 x + 1 4 dx = 5 3 . V arX = EX 2 - ( EX ) 2 = 5 3 - 49 36 = 11 36 . (d) Are the random variables X and Y independent? (1 points) f Y ( y ) = Z 2 x =0 1 8 ( x + y ) dx = y + 1 4 , 0 < y < 2 . Therefore, f ( x, y ) 6 = f X ( x ) f Y ( y ), that is, X and Y are NOT independent.
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Unformatted text preview: (e) What is the conditional pdf of X given Y = y ? (2 points) f X | Y = y = f ( x, y ) f Y ( y ) = 1 8 ( x + y ) 1 4 ( y + 1) = x + y 2( y + 1) , < x < 2 . (f) What is P ( X + Y > 1)? (1 points) P ( X + Y > 1) = 1-P ( X + Y ≤ 1) = 1-Z 1 x =0 ±Z 1-x y =0 1 8 ( x + y ) dy ² dx = 1-Z 1 x =0 1 8 ³ xy + y 2 2 ´µ µ µ µ 1-x y =0 dx = 1-Z 1 x =0 1 8 ³-1 2 x 2 + 1 2 ´ dx = 1-1 24 = 23 24 1...
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