EXAMPLE 5.19

Many students applying for college take the SAT, which for a few years consisted of three components: Critical Reading, Mathematics, and Writing. While some colleges used all three components to determine admission, many only looked at the first two (reading and math). Let $X$ and $Y$ denote the Critical Reading and Mathematics scores, respectively, for a randomly selected student. According to the College Board website, the population of students taking the exam in Fall 2012 had the following characteristics: ${\mu }_1=496,{\sigma }_1=114,{\mu }_2=514,{\sigma }_2=117$.

Suppose that $X$ and $Y$ have (approximately, since both variables are discrete) a bivariate normal distribution with correlation coefficient $\rho =.25$. The Matlab software package gives $P\left(X\le 650,Y\le 650\right)=P\left(\text{both scores are at most 650}\right)=.8097$.