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From: gilbert@turtle.DEC (Peter D Gilbert)
Newsgroups: net.math
Subject: conditional probability problem
Message-ID: <545@decwrl.UUCP>
Date: Thu, 24-May-84 17:26:25 EDT
Article-I.D.: decwrl.545
Posted: Thu May 24 17:26:25 1984
Date-Received: Thu, 31-May-84 19:35:25 EDT
Organization: DEC Engineering Network
Lines: 47

Re: Steve Vegdahl's conditional probability problem

	> Y and Z are independent random variables, uniform distribution on [0,1].
	> The random variable X is max(Y, Z).
	> 
	> Given that Y > 0.5, what is the expected value of X? the probability
	> distribution for X?
	>
	> Given that X > 0.5, what is the expected value of Y? the probability
	> distribution for Y?

Y and Z are independent random variables, uniform distribution on [0,1].
The random variable X is max(Y, Z).

In the following, it is assumed that 0 <= a,b <= 1.

The probability that X < a, given that Y > b is:

    P( X < a | Y > b )	= a(max(a,b)-b)/(1-b)

Other results:

    P( X < a | Y < b )	= a*min(a,b)

    P( Y < b | X > a )	= (b - a*min(a,b))/(1-a^2)

    P( Y < b | X < a )	= min(a,b)/a

    P( X < a )		= a^a

The expected value of X, given that Y > b is:

    E( X | Y > b )	= ( b + sqrt(b^2-2b+2) ) / 2

Other results:

    E( X | Y < b )	= 1/( 2 min(b,sqrt(1/2)) )

    E( Y | X > a )	= sqrt( (a^2+1)/2 )

    E( Y | X < a )	= a/2


		Peter D Gilbert
		Digital Equipment Corporation

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