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Let $Z_1$, $Z_2$, and $Z_3$ be independent normal random variables each having mean $0$ and variance $1$. If the random variables $X$ and $Y$ are defined by: $$X= \;\;\;\;\;\;\;\; Z_2-Z_3$$ $$Y=Z_1-Z_2+Z_3$$ Find the joint density function $f_{X,Y}$ of $X$ and $Y$ and hence the marginal density function $f_Y$ of $Y$.

My initial approach was to use the fact that the joint p.d.f. of $X$ and $Y$ is equal to the product of their marginal p.d.f.s however as I do not know the marginal p.d.f.s I am struggling to find another angle to approach the question. Any direction on how to approach the question or a solution would be much appreciated.

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  • $\begingroup$ You can use the following property: If $Z$ is a random vector in $\mathbb{R}^m$ with expected value $\mu\in\mathbb{R}^m$ and covariance matrix $\Gamma\in M_m(\mathbb{R})$, and if $A$ is any matrix of size $n\times m$, then the random vector $AZ$ in $\mathbb{R}^n$ has expected value $A\mu$ et covariance matrix $A\Gamma A^t$ . $\endgroup$ – Yoël Apr 27 '17 at 15:55
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    $\begingroup$ Since X and Y are both sum of independent normal variables , both will be Normal. Since cov(X,Y) is not equal to zero, the joint distribution of (X,Y) will be bivariate Normal. . $\endgroup$ – Croma14 Apr 27 '17 at 17:40

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