stevev@tekchips.UUCP (Steve Vegdahl) (04/26/84)
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? Steve Vegdahl Tektronix, Inc.
gilbert@turtle.DEC (Peter D Gilbert) (05/25/84)
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 ...{decvax,ucbvax,allegra}!decwrl!rhea!turtle!gilbert