For what value of $c$ is $f(x,y)$ a probability density function?

Question

I know that for answering this question I have to calculate the double integral of $f(x,y)=ce^{-2x^2-8y^2}$ and set it to $1$. Someone told me to use the gamma distribution function but I don’t know how.

Answer 1

This is a multivariate normal distribution with $\mathbf\mu=\mathbf0$ and $\mathbf\Sigma^{-1}=\begin{bmatrix}4&0\\0&16\end{bmatrix}$. Thus $$c=\frac1{\sqrt{(2\pi)^k\det\mathbf\Sigma}}=\frac1{2\pi\sqrt{1/64}}=\frac4\pi$$ where $k=2$ is the number of variables.