Probability that $X < Y$ for bivariate normal variable

Question

For the following probability questions, I'm a bit lost on how to proceed.

Let $X,Y$ be bivariate normal random variables with $\mathbb{E}(X)=10$, $\mathbb{E}(Y)=-7$, $Var(X)=30$, $Var(Y)=100$, and $\rho(X,Y)=-0.2$. Determine:

a) $P(X<\mathbb{E}(Y)-Y)$,

b) $P(X+10Y<0)$.

For a), I have been thinking about integrating the joint distribution to get the unconditional distribution of $X$ but since both R.V.'s are in the inequality this doesn't seem to be the way to go. Any ideas?

Answer 1

Hint: $X+Y$ and $X+10 Y$ are normal random variables. What are their means and variances?