Finding the distribution of X + Y using mean variance and correlation coefficient

Question

Suppose $(X,Y)$ has the bivariate normal distribution with mean $(0,1)$ and variance $(1,1)$ and correlation coefficient $.5$. What is the distribution of $X + Y$?

Now, I am in an accelerated statistics course. We tend to go over extremely difficult examples at a high speed, so trivial concepts fly by, and I tend to miss out on them. If someone could explain to me the process of answering this question, it would be helpful. I got normal distribution with mean 1 and variance 1 as the answer, but I am not confident in this.

Any help would be appreciated. Thank you.

Answer 1

• Distribution: linear combinations of the components of a bivariate distribution (e.g., $aX+bY$ for constants $a,b$) are Gaussian. (Check your definition of bivariate Gaussian.)
• Mean: how is $E[X+Y]$ related to $E[X]$ and $E[Y]$?
• Variance: how is $\text{Var}(X+Y)$ related to $\text{Var}(X)$, $\text{Var}(Y)$, and $\text{Cov}(X,Y)$? How is the covariance $\text{Cov}(X,Y)$ related to the correlation $0.5$?