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Unformatted text preview: Stats 116: HW5 August 2, 2007 16. (a) A happens iff at least one A i happens, so that A = ∪ n i =1 A i . (b) if A i happens, it means all points P j for j negationslash = i are to the left of P i going clockwise, or A i is to the right of P j , which contradict A j , i.e A i and A j are mutually exclusive. (c) Since the A i are mutually exclusive, P( A ) = n summationdisplay i =1 P( A i ) = n P( A 1 ) the last equality is by symmetry. Given P 1 , each other point has probability one half to fall in the left semicircle, so that P( A 1 ) = 1 2 n 1 , and P( A ) = n 1 2 n 1 20. (a) f ( x, y ) = g ( x ) h ( y ), with g ( x ) = xe x , and h ( y ) = e y , which both are proper densities, hence X and Y are independent. (b) P( Y ≤ . 5) > 0 and P( X > . 5) > 0, but P( X > . 5 , Y ≤ . 5) = 0 since X < Y . Then P( X > . 5 , Y ≤ . 5) negationslash = P( Y ≤ . 5)P( X > . 5) and X and Y are not independent 26. (a) Since A , B and C are independent, the joint cdf is just the product of the marginal cdfs...
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 Summer '07
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 Probability, Trigraph, ey dy dx

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