A problem in joint random variable (bivariate normal)

Suppose that $Y_{1}$ and $Y_{2}$ follow a bivariate normal distribution with parameters $\mu_{Y_{1}}=\mu_{Y_{2}}=0$, $\sigma^{2}_{Y_{1}}=1, \sigma^{2}_{Y_{2}}=2$, and $\rho=1/\sqrt{2}$. Find a linear transformation $x_1=a_{11}y_1+a_{12}y_2, x_2=a_{21}y_1+a_{22}y_2$ such that $x_1$ and $x_2$ are independent standard normal random variable.

I work on $f(y_1,y_2)=\frac{1}{2\pi\sigma_{Y_{1}}\sigma_{Y_{2}}\sqrt{1-\rho^2}}exp^(-\frac{1}{2(1-\rho^2)}[\frac{(y_1-\mu_{Y_{1}})^{2}}{\sigma^{2}_{Y_{1}}}+\frac{(y_2-\mu_{Y_{2}})^{2}}{\sigma^{2}_{Y_{2}}}-\frac{2\rho(y_1-\mu_{Y_{1}})(y_2-\mu_{Y_{2}})}{\sigma_{Y_{1}}\sigma_{Y_{2}}}]$

=> $f(y_1,y_2)=\frac{1}{2\sqrt{2}\pi}exp^(-({(y_1)}^{2}+(y_2)^{2}/2-\sqrt{2}(y_1)(y_2)/2)$

=> $f(x_1,x_2)=\frac{1}{2\pi}exp^{(-{(y_1)}^{2}/2-(y_2)^{2}/2)}$

Jacboian and some coefficient compairation, but I can only find $a_{11}a_{22}=2\sqrt{2}$, $a_{12}a_{21}=\sqrt{2}$, and fail in the comparaion...

Is there any simple way to do?

• All I get from my book is independent... – Richard Apr 7 '15 at 15:45

Let $X=(x_1,x_2)$ and $X=AY$ then $Y \sim N(0,A\Sigma_YA')$. The off-diagonal element of $A\Sigma_YA'$ is $$a_{12} (a_{11} + a_{21}) + (a_{11} + 2 a_{21}) a_{22}.$$
Now since $x_1$ and $x_2$ are jointly normally distributed, if they are uncorrelated then they will be independent too. Hence we have choose the parameters of transformation matrix $A$ such that the covariance is zero. For example let $a_{12}=x a_{21}$ and $a_{11}=a_{22}=1$ to have the covariance as $$x a_{21} (1 + a_{21}) + (1 + 2 a_{21})$$
Solving for $a_{21}$ in terms of $x$ you obtain$$a_{21}=\frac{-2-x-\sqrt{4+x^2}}{2x} \mbox{, or } a_{21}=\frac{-2-x+\sqrt{4+x^2}}{2x}$$
for any nonzero value for $x$ you obtain a covariance of zero, which implies independence (since we have jointly normally distributed $x_1$ and $x_2$). Further since the variance is $1$ each is marginally normal standard.
Yes, if $(Y_1,Y_2,\ldots,Y_n)$ has a multivariate distribution, then any $Y_i$ can be written as a linear combination of independent $N(0,1)$ random variables. You need the eigenvalues and eigenvectors of the covariance matrix. They can be explicitly given for the special case of $n=2$. A clear explanation can be found in Chapter 12 of the book Henk Tijms, Understanding Probability, 3rd ed.