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HW1solcorr - 9 That K n ∈ B R for all n implies that ∩...

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Unformatted text preview: 9. That K n ∈ B ( R ) for all n implies that ∩ n K n ∈ B ( R ). Note that { K n } is decreasing and λ ( K n ) = ( 2 3 ) n . Thus, by the continuity of measure, λ ( K ) = lim n →∞ λ ( K n ) = 0. 23. Note that P( Y ≤ y ) = Φ( y ∨ 1)- Φ(- 1) for y < 2Φ(- 1) + { Φ( y ∨ 1)- Φ(- 1) } otherwise . Thus, the density p ( y ) of Y w.r.t. λ + μ is p ( y ) = 2Φ(- 1)1 { y =0 } + φ ( y )1 { < | y | < 1 } . 30. a) For k ∈ N , P( X = k ) = P( k ≤ W < k + 1) = (1- e- λ ( k +1) )- (1- e- λk ) = e- λk (1- e- λ ) . b) P( Y ≤ y | X = k ) = P( Y ≤ y,X = k ) / P( X = k ) = P( k ≤ W < k + y ) / P( X = k ) = e- λk (1- e- λy ) e- λk (1- e- λ ) = 1- e- λy 1- e- λ . Since this doesn’t depend on k , Y is independent of X . Thus, for 0 < y < 1, P( Y ≤ y ) = P( Y ≤ y | X = k ) = 1- e- λy 1- e- λ . 1 c) E Y = Z 1 yλe- λy 1- e- λ dy = 1 1- e- λ- ye- λy- 1 λ e- λy 1 = 1 1- e- λ 1 λ- e- λ- 1 λ e- λ = 1 λ- e- λ 1- e- λ ....
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HW1solcorr - 9 That K n ∈ B R for all n implies that ∩...

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