History suggests that scores on the Math portion of the Stan

History suggests that scores on the Math portion of the Standard Achievement Test (SAT) are normally distributed with a mean of 529 and a variance of 5732. History also suggests that scores on the Verbal portion of the SAT are normally distributed with a mean of 474 and a variance of 6368. Select two students at random. Let X denote the first student\'s Math score, and let Y denote the second student\'s Verbal score. What is P(X > Y)?

Solution

X-Y follows Normal distribution with mean = 529-474 = 55 and

standard deviaton = sqrt(5732+6368) = 110

So the probability is

P(X > Y) = P((X-Y)>0) = P((X-Y-mean)/s >(0-55)/110)

=P(Z>-0.5) =0.6915 (from standard normal table)