Suppose (X) has a bivariate normal distribution, with mean 0 , (mathrm{V}left(X_{1} ight)=sigma_{1}^{2}), and (mathrm{V}left(X_{2} ight)=sigma_{2}^{2}). Let

Suppose \(X\) has a bivariate normal distribution, with mean 0 , \(\mathrm{V}\left(X_{1}\right)=\sigma_{1}^{2}\), and \(\mathrm{V}\left(X_{2}\right)=\sigma_{2}^{2}\). Let \(F_{1}\) and \(F_{2}\) be the CDFs of two continuous univariate distributions.

Now, let

\[ Y=\left(Y_{1}, Y_{2}\right)=\left(F_{1}^{-1}\left(\Phi\left(X_{1} / \sigma_{1}\right)\right), F_{2}^{-1}\left(\Phi\left(X_{2} / \sigma_{2}\right)\right)\right) \]

where \(\Phi\) is the CDF of a univariate \(\mathrm{N}(0,1)\) distribution.

Show that \(Y_{1}\) has a marginal distribution with \(\mathrm{CDF} F_{1}\).