Assume we have a large data set of PSAT and SAT scores with bivariate normal distribution with $\rho = 0.6$. The mean and SD of the PSAT scores are (respectively) $1200$ and $100$. The mean and SD of the SAT scores are (respectively) $1300$ and $90$. What percentage of students got at least 50 points more on the SAT than on the PSAT.
Let $X$ denote PSAT scores and $Y$ denote SAT scores. The desired probability is $$P(90Y + 1300>100X +1200+50).$$
I don't understand why it shouldn't be $P(Y>X +50)$, however.
Any help would be appreciated. Thanks.