Elisabeth Esparza

2022-07-18

Let ${Y}_{1}$ and ${Y}_{2}$ be independent random variables with ${Y}_{1}\sim N(1,3)$ and ${Y}_{2}\sim N(2,5)$ . If ${W}_{1}={Y}_{1}+2{Y}_{2}$ and ${W}_{2}=4{Y}_{1}-{Y}_{2}$ , what is the joint distribution of ${W}_{1}$ and ${W}_{2}$ ?

polishxcore5z

Beginner2022-07-19Added 14 answers

Step 1

First note that $({Y}_{1},{Y}_{2}{)}^{\prime}$ is bivariate normal because ${Y}_{1},{Y}_{2}$ are independent.

Next observe that

$\left(\begin{array}{c}{W}_{1}\\ {W}_{2}\end{array}\right)=A\left(\begin{array}{c}{Y}_{1}\\ {Y}_{2}\end{array}\right)$

$A=\left(\begin{array}{cc}1& 2\\ 4& -1\end{array}\right),$ ,

and use the fact that affine transforms of normal random vectors are normal, i.e.

$X\sim N(\mu ,\mathrm{\Sigma})\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}AX+b\sim N(A\mu +b,A\mathrm{\Sigma}{A}^{\prime}).$ .

First note that $({Y}_{1},{Y}_{2}{)}^{\prime}$ is bivariate normal because ${Y}_{1},{Y}_{2}$ are independent.

Next observe that

$\left(\begin{array}{c}{W}_{1}\\ {W}_{2}\end{array}\right)=A\left(\begin{array}{c}{Y}_{1}\\ {Y}_{2}\end{array}\right)$

$A=\left(\begin{array}{cc}1& 2\\ 4& -1\end{array}\right),$ ,

and use the fact that affine transforms of normal random vectors are normal, i.e.

$X\sim N(\mu ,\mathrm{\Sigma})\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}AX+b\sim N(A\mu +b,A\mathrm{\Sigma}{A}^{\prime}).$ .

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