Sum of normally distributed random variables is normal [duplicate]

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Given $(X,Y)$ normal vector with:$$ E\left(X\right)=0,\,E\left(Y\right)=0,\,Var\left(X\right)=4,\,Var\left(Y\right)=1 $$and $Cov\left(X,Y\right)=-1$, I was asked to to calculate $f_Z(2)$ where $Z=X+2Y$. In the solution they said that:$$ Z\sim N(E(X+2Y),Var(X+2Y)) $$It made me think - Is it true to say that the sum of two random variables $X$ and $Y$ that have normal distribution is also normal distribution? I know that if $X$ and $Y$ are independent that is true and even $X+Y\sim N(\mu_X+\mu_Y,\sigma_X^2+\sigma_Y^2)$ but can we says the $X+Y$ is normal if we don't know that they are independent? If not, how did they got $Z\sim N(E(X+2Y),Var(X+2Y))$? If yes, is there a general formula of $X+Y\sim N(\mu_X+\mu_Y,\sigma_X^2+\sigma_Y^2)$ or something similar?

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1 Answer

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$$X\pm Y \sim N(\mu_X\pm\mu_Y; \sigma_X^2+\sigma_Y^2 \pm 2Cov(X,Y))$$

anyway...you can have a look here

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