# Joint distribution of two Gaussian random variables

We have two independent Gaussian random variables with zero mean and variance $$\sigma^2$$, i.e., $$\theta_V \sim \mathcal{N}(0,\sigma^2)$$ and $$\theta_H \sim \mathcal{N}(0,\sigma^2)$$.

Let $$X=\theta_V^2+\theta_H^2$$ and $$Y=2\theta_V^2+2\theta_H^2+\alpha \theta_V$$, where $$\alpha$$ is a real number. How can we derive the joint probability density function (PDF) or cumulative density function (CDF) of $$X$$ and $$Y$$?

• It is impossible to do that unless we know a joint CDF for $\theta_V, \theta_H$. The easiest case is when they are independent. – Michael Apr 12 '19 at 13:24
• @Michael Those two variables are independent. Thank you for your kind reminding. – Hui Zhao Apr 12 '19 at 13:27
• One way is to use a Jacobian formula for the transformation $(V, H) \rightarrow (X,Y)$. Note that I am simplifying notation and using $V$ instead of $\theta_V$ and $H$ instead of $\theta_H$. – Michael Apr 12 '19 at 13:28
• @Michael I know Jacobian matrix is a valid way, but when we express $\theta_H$ in terms of $X$ and $Y$, there are two roots for $\theta_H$. I do not know how to solve it. – Hui Zhao Apr 12 '19 at 13:33
• For a transformation $(V,H)\rightarrow(X,Y)$, if there are $n$ distinct points $(v_i,h_i)$ that map to the same $(x,y)$ value then $$f_{X,Y}(x,y)=\sum_{i=1}^n f_{V,H}(v_i,h_i)|J(v_i,h_i)|^{-1}$$ – Michael Apr 12 '19 at 13:53

This answer summarizes my comments: One way is to use the Jacobian matrix method. For a transformation $$(V,H)\rightarrow(X,Y)$$, if there are $$n$$ distinct points $$(v_i,h_i)$$ that map to the same $$(x,y)$$ value then $$f_{X,Y}(x,y) = \sum_{i=1}^n f_{V,H}(v_i,h_i)|J(v_i,h_i)|^{-1}$$ where $$|J(v_i,h_i)|$$ is the determinant of the $$2 \times 2$$ Jacobian matrix for the point $$(v_i,h_i)$$: $$J(v_i,h_i) = \left[\begin{array}{}dx/dv \quad dx/dh\\ dy/dv \quad dy/dh\end{array}\right]\left|_{(v_i,h_i)}\right.$$
In this particular case we have: $$f_{V,H}(v,h) = \frac{1}{2\pi \sigma^2}e^{-\frac{(v^2+h^2)}{2\sigma^2}} \quad \forall (v,h) \in \mathbb{R}^2$$ Since $$X=V^2+H^2$$ and $$Y=2V^2+2H^2 + \alpha V$$, we can compute the Jacobian to get $$|J(v,h)|^{-1} = 1/|2\alpha h|$$, and for $$(x,y)$$ values with $$n=2$$ we get: $$f_{X,Y}(x,y) = \sum_{i=1}^2 \frac{f_{V,H}(v_i,h_i)}{2|\alpha h_i|}$$ where it suffices to solve for the corresonding $$(v_1, h_1)$$ and $$(v_2, h_2)$$ points.