# How to create two independent exponential distributions from two arbitrary exponential distributions.

In the normal case, we know that if an $$n-$$dimensional random variable $$X$$ has a multivariate normal distribution with mean vector $$\mu\in \mathbb{R}^n$$ and covariance matrix $$C\in \mathbb{R}^n \times \mathbb{R}^n$$, i.e., $$X\sim \mathcal{N}(\mu,C)$$, then $$Z=C^{-1/2}(X-\mu)\sim\mathcal{N}(0,I_n),$$ that is, every component r.v. $$Z_i$$ of the vector $$Z$$ now has a standard normal distribution and each $$Z_i$$ is independent of $$Z_j$$, for $$i\neq j$$.

I just want to ask: is there already a result like this for other (multivariate) distributions? For example, given $$X_1$$ and $$X_2$$ form a bivariate exponential distribution with covariance $$C$$, can we form random variables $$Y_1, Y_2$$, still forming a bivariate exponential distribution but are now independent?