Calculating Covariance and Variance

Question

We have $E(X) = 10,$ $\mathrm{Var}(X) = 15,$ $E(Y) = 4,$ $\mathrm{Var}(Y ) = 12,$ and $\mathrm{Corr}(X, Y ) = 0.25$.

My aim is to find

1. $\mathrm{Var} (X+Y)$, and
2. $\mathrm{Cov}(X, X+Y)$

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For 1, I started with $\mathrm{Var} (X+Y) = \mathrm{Var}(X) + \mathrm{Var} (Y) + 2\mathrm{Cov}(X,Y)$ and I tried to calculate the last term by calculating $\mathrm{Corr}(X,Y)$ multiply $\sqrt{\mathrm{Var}(X)\;\mathrm{Var}(Y)}$.

For 2, I started with $\mathrm{Cov}(X, X+Y) = \mathrm{Var} (X) + \mathrm{Cov}(X,Y)$ which can be found as above.

Just checking is my approach correct? Any help will be appreciated!

Answer 1

$\text{Cov}(X,X+Y)=E[X(X+Y)]-E(X)E(X+Y)=E(X^2+XY)-E(X)[E(X)+E(Y)]=E(X^2)+E(XY)-[E(X)]^2-E(X)E(Y)=\text{Var}(X)+\text{Cov}(X,Y)$.

So yes, you are doing (b) correctly.