(X, Y) Bivariate normally distribution

Question

If $(X, Y)$ is bivariate normally distributed with marginals means $1$, marginal variances $1$ and correlation coefficient $\rho=0.2$, what is the distribution of $X + Y$ ? How can I start the problem ?

Answer 1

For each measurable function g for a given rv $(X,Y)$ with density $f_{(X,Y)}$ we can calculate $E[g(X,Y)]$ by $$\int_{-\infty}^\infty\int_{-\infty}^\infty g(x,y) f_{(X,Y)}(x,y) dx\, dy$$

Also it holds for a random variable X that the cdf can be calculated by $$F_X(x) = P(X \le x) = E[1_{X\le x}]$$

So it follows:

$$P(X + y \le z) = E[1_{X + Y \le z}] = \int_{-\infty}^\infty\int_{-\infty}^{z-y} f_{(X,Y)}(x,y) dx\, dy$$

by setting $g(x,y) = 1_{X + Y \le z}$

And the density of $(X,Y)$ is well known…