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If $X \sim N(0, \sigma_1^2)$ and $Y \sim N(0, \sigma_2^2)$ are independent random variables, then the joint pdf of $(X,Y)$ is say $f(x,y)$: Given $Z = \sqrt{X^2 + Y^2}$, you seek $\text{Var}(Z)$: …

Then, define a copula function, which is a function of the two cdf's $F$ and $G$ that creates a bivariate joint distribution function (cdf) from $F$ and $G$, such that the marginal pdf's of $X$ and $Y$ … Here, I use a Morgenstern copula with parameter $\alpha$ that induces correlation (there are many many other Copula functions available): Let $h(x,y)$ denote the bivariate Triangular joint pdf obtained …

OP wrote: I've got 2 gamma-distributed random variables (X,Y) with ... say $Y=\sqrt{X}$. Your question is internally inconsistent. In particular, if $X$~Gamma$(a,b)$ with pdf $f(x)$, say: $$f(x) …

It depends what one means by simple ... Given $\mu = (0,0)$, $\quad \Sigma =\left( \begin{array}{cc} \sigma _1^2 & \rho \sigma _1 \sigma _2 \\ \rho \sigma _1 \sigma _2 & \sigma _2^2 \\ \end{arr …