3225_hw1 - Homework 1, Math 3225 September 10, 2010 1....

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Unformatted text preview: Homework 1, Math 3225 September 10, 2010 1. Recall that events A 1 , ..., A k are independent if for every non-empty subset S of { 1 , ..., k } we have P ( s S A s ) = productdisplay s S P ( A s ) . (1) As a consequence of this, it turns out that this implies, and is equivalent to, the statement Claim. For i = 1 , 2 ..., k we have that for any set B in the -algebra generated by all the sets A j , j negationslash = i , P ( A i , B ) = P ( A i ) P ( B ) . In the special case i = 1 this would be saying that for any B ( A 2 , ..., A k ) i.e. B is any set gotten by doing any number of in- tersections, unions and complements (which will turn out to be finite in number, since k < ) of the events A 2 , ..., A k we must have that P ( A 1 , B ) = P ( A 1 ) P ( B ). This equivalence is somewhat tricky to prove; and to give you a taste of what is involved, your first problem will be to prove the following: Suppose that A, B, C, and D are independent events (using the (1)...
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3225_hw1 - Homework 1, Math 3225 September 10, 2010 1....

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