Joint normal random variables, covariance, and probability

Question

I'm having a lot of trouble with this question:

X and Y are joint normal random variables with common mean 0, common variance 1, and covariance 1/2. What is $P(X+Y\leq \sqrt{3})$?

Thank you!

Answer 1

if they are jointly normal, then $Z=X+Y$ is also normal with mean 0 and variance $$var(Z)=E[(Z-0)]^2=E[Z^2]=E[(X+Y)^2]$$ $$var(Z)=E[X^2+Y^2+2XY]=1+1+2\cdot \frac{1}{2}=3$$ because $cov(X,Y)=E[XY]-E[X]E[Y]=0.5$.

So we have that $Z\sim \mathcal{N}(0,3)$, and $P(Z\leq \sqrt 3)$ is

$$P(Z\leq \sqrt 3)=P(Z/\sqrt{3}\leq \sqrt 3 / \sqrt{3})=P(A\leq 1)$$ where $A\sim \mathcal{N}(0,1)$, so the final result is $\Phi(1)$.