How to calculate covariance of X and Y given joint probability

Question

$X$ and $Y$ are dependent variables both normally distributed as $N(\mu-const, \sigma^2)$. I don't know what the joint distribution is.

I know that when $const = 0$, then the joint probability yields: $P(X < 0, Y < 0) = 1/6$

Now I want to calculate $P(X < 0, Y < 0)$ for different values of $const<0$. For this I attempted to compute the multivariate normal distribution but:

1. I don't know the value of $cov(X,Y)$.
2. I don't know the value of $V(X+Y)$.

So I was wondering if, knowing the value of $P(X < 0, Y < 0)$ for $const=0$, I could derive the value of $cov(X,Y)$ so I could finally compute $P(X < 0, Y < 0)$ for other values of $const<0$.

Any tips?

Answer 1

It is immediately obvious that there is insufficient information to obtain the desired probability: you have, for a given constant, three parameters that may vary: the common marginal mean $\mu$, the common marginal variance $\sigma^2$, and the correlation $\rho$; yet you have only one condition which is that when the constant is zero, $$\Pr[X < 0 \cap Y < 0] = 1/6.$$ You can create an infinite subfamily of normal distributions with a variety of such parameters that satisfies the probability constraint.