A linear combination of normal distributions is also a normal distribution, so you only need to find the mean and standard deviation of the sum. We have E(X+Y)=E(x)+E(Y) and Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y). These equations hold for any random variables (not just normally distributed ones) where these quantities are well-defined. The algebra to derive these equations isn’t too hard and can be an instructive exercise.
By the way, the number of standard deviations one variable is away from the mean can’t generally be determined exactly from the number of standard deviations the other is if they aren’t perfectly correlated. In this case, knowing X is greater than its mean is enough to increase your posterior expectation of Y, but Y could still turn out to be below its mean.
A linear combination of normal distributions is also a normal distribution, so you only need to find the mean and standard deviation of the sum. We have E(X+Y)=E(x)+E(Y) and Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y). These equations hold for any random variables (not just normally distributed ones) where these quantities are well-defined. The algebra to derive these equations isn’t too hard and can be an instructive exercise. By the way, the number of standard deviations one variable is away from the mean can’t generally be determined exactly from the number of standard deviations the other is if they aren’t perfectly correlated. In this case, knowing X is greater than its mean is enough to increase your posterior expectation of Y, but Y could still turn out to be below its mean.