helpsession5 - AMS 507 Recitation 5 Xiaoping Zhou November...

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Unformatted text preview: AMS 507 Recitation 5 Xiaoping Zhou November 15, 2010 Problem 7.10 Important note: don't take for granted that the trials are independent, oth- erwise P{X=3} is a constant. (a) Let X i = ( 1 , if the i-th trial succeeds , if the i-th trail fails , then EX i = P ( X i = 1) , E [ 3 X i =1 X i ] = 1 . 8 . As the three trails have same probability to succeed, then EX i = 0 . 6 , i = 1 , 2 , 3 . P { X = 3 } = P { X 1 = 1 , X 2 = 1 , X 3 = 1 } P { X 1 = 1 } = 0 . 6 If they are independent, we would know that P { X = 3 } = P { X 1 = 1 } P { X 2 = 1 } P { X 3 = 1 } = 0 . 6 3 . The largest probability that all of them succeed will not exceed the prob- ability that each trial to succeed. Let us de ne a case that when the three trials are perfectly dependent (i.e, as soon as one of them succeeds, so do the other two.) As an example, we de ne a uniform random variable U ~U(0,1), and let X i = ( 1 , if U . 6 , o.w , i = 1 , 2 , 3 ....
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helpsession5 - AMS 507 Recitation 5 Xiaoping Zhou November...

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