# NotesNov5 - The Law of Total Probability in the context of...

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The Law of Total Probability in the context of random variables Let X 1 ,X 2 ,...,X n be random variables; some of the X i s may be discrete, and some other continuous; let A be some event expressed in terms of X 1 ,...,X n . The law of total probability can be used to compute the probability of A by conditioning on the value of one (or more) of the X s.

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Conditioning on a discrete random variable Suppose that one of the random variables, X i , is discrete with a pmf p X i . Then P ( A ) = X x i p X i ( x i ) P ( A | X i = x i ) .
Conditioning on a continuous random variable Suppose that one of the random variables, X i , is continuous, with a pdf f X i . Then P ( A ) = Z -∞ f X i ( x i ) P ( A | X i = x i ) dx i .

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Example : Let X 1 ,X 2 ,...,X n be independent exponential random variables, X i Exp( λ i ) , i = 1 , 2 ,...,n . Compute the probability that X i is the smallest among X 1 ,X 2 ,...,X n .
Bivariate Normal distribution The normal distribution is the single most

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## This note was uploaded on 12/03/2010 for the course OR 3500 at Cornell University (Engineering School).

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NotesNov5 - The Law of Total Probability in the context of...

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