HW+4+Solutions

# HW+4+Solutions - ISyE 2027 Fall 2011 Homework 4 Solutions 1...

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Unformatted text preview: ISyE 2027 Fall 2011 Homework 4 Solutions 1. a. X β‘ the number or hearts in a hand of 6 cards P ( X = k ) = ( 13 k )( 39 6 β k ) ( 52 6 ) k = 0 , 1 , 2 , . . . , 6 o/w. b. P ( X = i, Y = j ) = ( 13 i )( 13 j )( 26 6 β i β j ) ( 52 6 ) i, j β { , 1 , 2 , . . . , 6 } , i + j β€ 6 o/w. 2. a. P (β¦) = 1 , β c ( β 2) 2 + c ( β 1) 2 + c (1) 2 + c (2) 2 = 1 c (4 + 1 + 1 + 4) = 1 c = 1 10 b. P [ X = β 1] = 1 10 ( β 1) 2 = 1 10 c. E ( X ) = k =2 β k = β 2 k P ( X = k ) = 1 10 k =2 β k = β 2 k 3 = 1 10 ( β 8 β 1 + 1 + 8) = 0 d. E ( X 2 ) = k =2 β k = β 2 k 2 P ( X = k ) = 1 10 k =2 β k = β 2 k 4 = 1 10 (16 + 1 + 1 + 16) = 34 10 e. Let q be a median of X , then q satisfies the inequalities P ( X β€ q ) β₯ 1 2 and P ( X β₯ q ) β₯ 1 2 . This is true for q β [ β 1 , 1] f. From the (d) part, E ( X 2 ) = 34 10 ΜΈ = E ( X ) 2 = 0 3. a. P ( I A = k ) = { P ( A c ) k = 0 , P ( A ) k = 1 ....
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## This note was uploaded on 01/22/2012 for the course ISYE 2027 taught by Professor Zahrn during the Fall '08 term at Georgia Tech.

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HW+4+Solutions - ISyE 2027 Fall 2011 Homework 4 Solutions 1...

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