Sheet3 - (v) Could you Fnd P ( X + Y ) without knowing that...

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Statistics 330/600 Due: Thursday 3 February 2005: Sheet 3 Please attempt at least the starred problems. *(3.1) (separation of pairs of points) UGMTP Problem 2.5. *(3.2) Suppose X and Y are independent random variables on (Ä, F , P ) . That is, P { X A , Y B }= P { X A } P { Y B } for all A , B B ( R ) . Suppose also that each random variable has a Bin ( 2 , p ) distribution, for some Fxed p in ( 0 , 1 ) . (i) DeFne T : Ä R 2 by T (ω) = ( X (ω), Y (ω) ) . ±ind the distribution, P ,o f T under P . (That is, P is the image of P under T ). Hint: The distribution should be a probability measure concentrated on nine points in R 2 . (ii) DeFne ψ : R 2 R by ψ( x , y ) = x + y . ±ind the distribution, µ ,o f ψ under P . (iii) DeFne S : Ä R by S (ω) = X (ω) + Y (ω) . ±ind the distribution, Q ,o f S under P . (iv) Calculate P ( X +
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Unformatted text preview: (v) Could you Fnd P ( X + Y ) without knowing that X and Y are independent? (vi) ind the distribution of XY . ind P ( XY ) . (No cheating by using facts about independence that we have not yet established.) ind P ( X ) P ( Y ) . (vii) Could you Fnd P ( XY ) without knowing that X and Y are independent? (3.3) (general Borel-Cantelli converse) UGMTP Problem 2.2. You should interpret the assertion k X n / n k 2 1 to mean that P X 2 n / 2 n 1 as n . (3.4) (inner and outer regularity) UGMTP Problem 2.12. Note the typo in the deFnition of outer regularity: the F should be G ....
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