Let $F(\cdot,\cdot, \mu_1, \mu_2,\sigma_1^2,\sigma_2^2,\rho)$ denote the d.f. $(X,Y)$. Show that

$$\Bigg(\frac{X-\mu_1}{\sigma_1}, \frac{Y - \mu_2}{\sigma_2}\Bigg)$$

has a $N(0,0,1,1,\rho)$ distribution and, hence, express $F(\cdot,\cdot, \mu_1, \mu_2,\sigma_1^2,\sigma_2^2,\rho)$ in terms of $F(\cdot,\cdot, 0,0,1,1,\rho)$.

I know how to show the first part, but I am confused about the second part, how to express the relationship between the two? Any hint, advice or suggestion is appreciated.


1 Answer 1


You have done all the hard work already.

$$\begin{align} F(x,y; \mu_X,\mu_Y, \sigma_X^2, \sigma_Y^2, \rho) &= P\{X \leq x, Y\leq y\}\\ &= P\left\{\frac{X-\mu_X}{\sigma_X}\leq \frac{x-\mu_X}{\sigma_X}, \frac{Y-\mu_Y}{\sigma_Y}\leq \frac{y-\mu_Y}{\sigma_Y}\right\}\\ &= F\left(\frac{x-\mu_X}{\sigma_X},\frac{y-\mu_Y}{\sigma_Y}, 0, 0, 1^2, 1^2, \rho\right). \end{align}$$


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