#Variable 1 = SAT score
#Variable 2 = college GPA

mu1 <- 1040
mu2 <- 3.02

sigma1 <- 153
sigma2 <- 0.58
rho <- 0.35

library(mvtnorm)

mu.vec=c(mu1,mu2)
covar <- sigma1*sigma2*rho
covmat=matrix(c(sigma1^2,covar,covar,sigma2^2),nrow=2,byrow=T)

# What is the probability that a randomly selected student has 
# an SAT of less than 1000 and a GPA of less than 2.7?

pmvnorm(lower=c(-Inf,-Inf),
        upper=c(1000,2.7),
        mean=mu.vec,
        sigma=covmat)[1]

# What is the probability that a randomly selected student has 
# an SAT of more than 1100 and a GPA of less than 2.7?

pmvnorm(lower=c(1100,-Inf),
        upper=c(Inf,2.7),
        mean=mu.vec,
        sigma=covmat)[1]

# What is the probability that a randomly selected student has 
# an SAT of between 900 and 1000 and a GPA of between 3.6 and 3.9?

pmvnorm(lower=c(900,3.6),
        upper=c(1000,3.9),
        mean=mu.vec,
        sigma=covmat)[1]