I have three independent random variables, $x_1, x_2$ and $x_3$ with $x_1,x_2\cong N(0,1)$ and $x_3 \cong N(0, 1/2)$. I have $s:= \sqrt((x_1-x_2)^2 + 4x_3^2)$. How do I write down an integral for the pdf of $s$?
Asked differently, how do I get from step (2.1) to (2.2) in the following screenshot. The best I can guess is $\delta$ is the discontinuous square bump function of height $\epsilon/2 >0$ that integrates to 1 but I still don't see which law of probability is being used. (Let alone, why is it definitional)
The machinery I was looking for on delta functions is described here with an example probability distribution calculation https://ieeexplore.ieee.org/document/9418536