How to calculate the joint probability from two normal distributions

Question

I have two random variables $X$ and $Y$ both normally distributed as $N(\mu, \sigma^2)$ (they have the same distribution).

$X$ and $Y$ are dependent. They are defined from other random variables A, B and C also with gaussian distribution: $X$ = $A - B + const$ and $Y$ = $-A + C + const$.

A, B and C are independent and also equally distributed as $N(0, \sigma_2^2)$.

I want to calculate $P(X < x, Y < y)$.

How can I do this?

Answer 1

Is easy to see that $X$ and $Y$ are jointly gaussian; every linear combination of $X$ and $Y$ is in fact normal, thanks to the fact that $A$ $B$ and $C$ are independent.

Note also that the const you've been writing in the definition of $X$ and $Y$ is actually $\mu$.

Anyhow, you just need to calculate the covariance between $X$ and $Y$ (which is pretty easy to do) and you'll have a multivariate normal random variable, for which we know a lot (its density, for example).

On the other hand, just like in the case of a normal in one dimension, is not possible to find an explicit formula for $P(X < x, Y < y)$

Answer 2

HINT: If $X\sim N(\mu_x,\sigma^2_x)$ and $Y\sim N(\mu_y,\sigma^2_y)$ then $$X+Y\sim N(\mu_x+\mu_y,\sigma^2_x+\sigma^2_y)$$ $$X-Y\sim N(\mu_x-\mu_y,\sigma^2_x+\sigma^2_y)$$ $$X+const\sim N(\mu_x+const,\sigma^2_x)$$