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solution2

# solution2 - EL630 Homework2 Solutions Problem 1 solution...

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EL630: Homework2 Solutions Problem 1 solution – the table below lists all the outcomes and their probabilities. A 1 in a slot means that switch is closed and a 0 means that switch is open. outcome S1 S2 S3 S4 Probability 1 ξ 0 0 0 0 4 ) 1 ( p - 2 ξ 0 0 0 1 3 ) 1 ( p p - 3 ξ 0 0 1 0 3 ) 1 ( p p - 4 ξ 0 0 1 1 2 2 ) 1 ( p p - 5 ξ 0 1 0 0 3 ) 1 ( p p - 6 ξ 0 1 0 1 2 2 ) 1 ( p p - 7 ξ 0 1 1 0 2 2 ) 1 ( p p - 8 ξ 0 1 1 1 ) 1 ( 3 p p - 9 ξ 1 0 0 0 3 ) 1 ( p p - 10 ξ 1 0 0 1 2 2 ) 1 ( p p - 11 ξ 1 0 1 0 2 2 ) 1 ( p p - 12 ξ 1 0 1 1 ) 1 ( 3 p p - 3 ξ 1 1 0 0 2 2 ) 1 ( p p - 4 ξ 1 1 0 1 ) 1 ( 3 p p - 15 ξ 1 1 1 0 ) 1 ( 3 p p - 16 ξ 1 1 1 1 4 p Let = A {Signal is received at output}, = i S {i th switch is closed}. a. We can see that { } 16 15 14 13 12 11 10 9 8 7 6 , , , , , , , , , , ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ = A . To find ) ( A P we just add up the probabilities of the outcomes in A , to get 4 3 2 3 2 2 3 4 3 2 ) 1 ( ) 1 ( 5 ) 1 ( 4 ) ( p p p p p p p p p p p A P + - + = - + - + - + = . b. We want to find ) ( 4 A S P , which can be written as

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) ( ) ( ) ( 4 4 A P A S P A S P = . Now, we can see, from the table above that { } 15 13 11 9 7 4 , , , , ξ ξ ξ ξ ξ = A S and the probability of this event is found by adding up the probabilities of the outcomes in A S
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solution2 - EL630 Homework2 Solutions Problem 1 solution...

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