solution5

# solution5 - EL630 Homework5 Solutions 1 a 5 3 5 2 5 1 1 1 1...

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Unformatted text preview: EL630: Homework5 Solutions 1) a) 5 / 3 5 / 2 5 / 1 ) 1 ( ) 1 ( }) 1 { } 1 ({ ) 1 ( 2 = + =- < + =- < = X P X P X X P X P P . b) 5 / 2 5 / 1 5 / 1 }) 1 { } 1 2 ({ ) ) (sin( = + = ≤ ≤- ≤ ≤- = ≥ X X P X P P π . 2) We will use the result (obtained by integration by parts k times) that 1 1 ) 1 ( 1 + =- - ∫ n dp p p k n k n k . a) Let = T {Tails}. 2 / 1 ) 1 ( ) 1 ( ) ( ) ( ) ( 1 1 =- = = = ∫ ∫ dp p dp p f p X T P T P X . b) We need to find ) ( B x f B X , where = B {six heads out of ten tosses}. ) ( ) ( ) ( ) ( B P p f p X B P B x f X B X = = . Now, 4 6 ) 1 ( 6 10 ) ( p p p X B P- = = 11 / 1 ) 1 ( ) 1 ( 6 10 ) ( ) ( ) ( 1 4 6 1 =- = = = ∫ ∫ dp p p dp p f p X B P B P X . So, 4 6 ) 1 ( 6 10 11 ) ( p p B X f B X- = . and ∫- = < < 75 . 35 . 4 6 ) 1 ( ) 1 ( 6 10 11 ) 75 . 35 . ( dp p p X P 3) a) ) ( ) ( ) ( ) ( ) ( 3 / 1 3 / 1 3 y F y X P y X P y Y P y F X Y = ≤ = ≤ = ≤ = ) ( 3 1 3 1 ) ( ) ( ) ( 3 / 1 3 / 2 3 / 2 3 / 1 3 / 1 y U e y y y f y F dy d y f y X X Y α α-- = = = . b) Here ) ( X g Y = , where 3 2 ) ( + = x x g . Then 2 ) ( = ′ x g , and there is only one solution for y in the equation 3 2 + = x y . It is 2 3- = y x . So ( 29 3 2 1 2 3 2...
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solution5 - EL630 Homework5 Solutions 1 a 5 3 5 2 5 1 1 1 1...

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