How to find the pdf of the difference beween random variables Let $X_1,X_2,...,X_n$ be realizations of a random variable $X_0$. 
Let $Y_1,Y_2,...,Y_n$ be realizations of a random variable $Y_0 \neq X_0$.   The distribution of $X_0$ and $Y_0$ is $N(0,\sigma^2_x)$ and $N(0,\sigma^2_y)$ and they are independent of each other. How can I find the pdf of the euclidean distance /difference between the two initial conditions i.e., $Z = {||X_0 - Y_0||}_2$
 A: As you stated in a comment, $X_0$ and $Y_0$ are independent, therefore the column vector $V = [X_0,Y_0]$ is Bivariate Normal with mean being the zero vector, and covariance matrix being the diagonal matrix with $\sigma^2_x$ and $\sigma^2_y$ on the diagonal.
$X_0 -Y_0 = BV$, where B is the row vector [1 -1]. Applying the properties of an affine transformation of a Multivariate Normal https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Affine_transformation , results in $X_0 - Y_0$ being $N(0,\sigma^2_x$ + $\sigma^2_y)$.
If you want to just understand more simply what is going on with the variance, then look at the properties of Variance provided at https://en.wikipedia.org/wiki/Variance#Properties . In particular, if $X$ and $Y$ are random variables and $a$ and $b$ are constants, then $$\operatorname{Var}(aX + bY) = a^2\operatorname{Var}(x) + b^2\operatorname{Var}(Y) + 2ab\operatorname{Cov}(X,Y).$$In this case, use $a = 1, b = -1$.
Finally, note that $Z = \operatorname{abs}(X_0 - Y_0)$, and therefore has the Half-Normal distribution http://en.wikipedia.org/wiki/Half-normal_distribution with parameter $\sigma^2_x$ + $\sigma^2_y$.
