Assignment3-Solutions

# Assignment3-Solutions - 1 Assignment 3, STAT220-Fall 2010...

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1 Assignment 3, STAT220-Fall 2010 Instructor: Kamyar Moshksar Due on Friday, October 29th Problem 1 - Let X Bi(10 , 0 . 1) and Y Bi(5 , 0 . 2) be two independent random variables deﬁned on similar sample spaces. Find P(X < Y) and P(X > Y | Y 2) . Solution- (a)- To compute P(X < Y) , we proceed as follows: P(X < Y) ( a ) = 5 X y =0 P(X < Y | Y = y ) P (Y = y ) = 5 X y =0 P(X < y | Y = y ) P (Y = y ) ( b ) = 5 X y =0 P(X < y ) P (Y = y ) ( c ) = 5 X y =1 P(X < y ) f Y ( y ) . (1) In this expression, ( a ) is by the law of total probability, ( b ) is by independence of X and Y and ( c ) is by the fact that P( X < 0) = 0 . Now, P( X < y ) = y - 1 X x =0 f X ( x ) . (2) By the above two expressions, P(X < Y) = 5 X y =1 y - 1 X x =0 f X ( x ) f Y ( y ) = 5 X y =1 y - 1 X x =0 ± 10 x ² (0 . 1) x (0 . 9) 10 - x ± 5 y ² (0 . 2) y (0 . 8) 5 - y . (3)

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2 (b)- To compute P(X > Y | Y 2) , P(X > Y | Y 2) = P(X > Y , Y 2) P(Y 2) . (4) To compute the numerator, P(X > Y , Y 2) = 10 X x =0 P(X > Y , Y 2 | X = x )P(X = x ) = 10 X x =0 P(X > Y , Y 2 | X = x ) f X ( x ) = 10 X x =0 P( x > Y , Y 2 | X = x ) f X ( x ) = 10 X x =0 P(2 Y < x ) f X ( x ) = 10 X x =3 P(2 Y < min { x, 6 } ) f X ( x
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## This note was uploaded on 01/26/2012 for the course ECON 401 taught by Professor Burbidge,john during the Fall '08 term at Waterloo.

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Assignment3-Solutions - 1 Assignment 3, STAT220-Fall 2010...

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