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The exercise is the following. Let $(X,Y)$ be a bi-variate normal vector with $\mu=(2,3)$ and covariance matrix $$\Sigma=\begin{pmatrix}1 & -2 \\ -2 & 4\end{pmatrix}.$$ What is the distribution of the random variable $X+Y$?

My try is: Consider the moment generating functions of $X$ and $Y$. Since $\sigma_X^2=1$, $\sigma_Y^2=4$, $\mu_X=2$ and $\mu_Y=3$, we have $$\varphi_X(t)=\exp(2t+\frac{t^2}{2})$$ $$\varphi_Y(t)=\exp(3t+2t^2).$$

Thus, $$\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)=exp(5t+\frac{5t^2}{2}).$$

Hence, $X+Y\sim N(5,5)$ since the moment generating function of a normal r.v. is given by $\varphi_X(t)=\exp(\mu_X t + \frac{\sigma_X^2 t^2}{2})$.

I am asking this because my teacher put as a solution $X+Y\sim N(5,1)$. Any feedback?

EDIT:

I think that we can not use $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ since $X$, $Y$ are not independent.

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2 Answers 2

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The variance of $X+Y$ is $$ \operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y)=1+4+2\times(-2)=1. $$ So $N(5,1)$ is correct. Your problem is the assertion $\varphi_{X+Y}(t)=\varphi_X(t)\varphi_Y(t)$ which doesn't work in this case because $X$ and $Y$ are dependent.

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The two marginal distributions are clearly non independent which means you have to find the sum of two non i.i.d. gaussians.

$X+Y \sim N(\mu_x + \mu_y, \sigma_x^2 + \sigma_y^2 +2 \sigma_{x,y})$

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