HW1solcorr - 9. That K n B ( R ) for all n implies that n K...

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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 ( K n ) = ( 2 3 ) n . Thus, by the continuity of measure, ( K ) = lim n ( K n ) = 0. 23. Note that P( Y y ) = ( y 1)- (- 1) for y < 2(- 1) + { ( y 1)- (- 1) } otherwise . Thus, the density p ( y ) of Y w.r.t. + is p ( y ) = 2(- 1)1 { y =0 } + ( y )1 { < | y | < 1 } . 30. a) For k N , P( X = k ) = P( k W < k + 1) = (1- e- ( k +1) )- (1- e- k ) = e- k (1- e- ) . b) P( Y y | X = k ) = P( Y y,X = k ) / P( X = k ) = P( k W < k + y ) / P( X = k ) = e- k (1- e- y ) e- k (1- e- ) = 1- e- y 1- e- . Since this doesnt depend on k , Y is independent of X . Thus, for 0 < y < 1, P( Y y ) = P( Y y | X = k ) = 1- e- y 1- e- . 1 c) E Y = Z 1 ye- y 1- e- dy = 1 1- e- - ye- y- 1 e- y 1 = 1 1- e- 1 - e- - 1 e- = 1 - e- 1- e- ....
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HW1solcorr - 9. That K n B ( R ) for all n implies that n K...

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