W8-slides1 - Lecture 22 Outline and Examples Independent...

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Lecture 22- Outline and Examples Independent random variables (Ross § 6.2) Sums of independent random variables (Ross § 6.3)
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Independent random variables. (Ross § 6.2) Example A. A coin is tossed once and heads turns up with probability p . Let X and Y be the number of heads and tails respectively. Are X and Y independent? Solution.
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Independent random variables. (Ross § 6.2) Example A. A coin is tossed once and heads turns up with probability p . Let X and Y be the number of heads and tails respectively. Are X and Y independent? Solution. No:
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Independent random variables. (Ross § 6.2) Example A. A coin is tossed once and heads turns up with probability p . Let X and Y be the number of heads and tails respectively. Are X and Y independent? Solution. No: P ( X = 1 , Y = 1) = 0 (in one toss we get either one head or one tail)
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Independent random variables. (Ross § 6.2) Example A. A coin is tossed once and heads turns up with probability p . Let X and Y be the number of heads and tails respectively. Are X and Y independent? Solution. No: P ( X = 1 , Y = 1) = 0 (in one toss we get either one head or one tail) P ( X = 1) P ( Y = 1) = p (1 - p ).
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Independent random variables. (Ross § 6.2) Example A. A coin is tossed once and heads turns up with probability p . Let X and Y be the number of heads and tails respectively. Are X and Y independent? Solution. No: P ( X = 1 , Y = 1) = 0 (in one toss we get either one head or one tail) P ( X = 1) P ( Y = 1) = p (1 - p ). Therefore X and Y are not independent since P ( X = 1 , Y = 1) 6 = P ( X = 1) P ( Y = 1).
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The joint pdf of two r.v.s X and Y is given by f X , Y ( x , y ) = ± (1 + x + y ) / 2 if 0 < x < 1 , 0 < y < 1 0 elsewhere Are X and Y independent? Solution.
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W8-slides1 - Lecture 22 Outline and Examples Independent...

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