How to calulate entropy $H(X+S+Z,X+\alpha S)$

Question

I have no idea how to calulate entropy $H(X+S+Z,X+\alpha S)$ where $X\sim N(0,P)$, $S\sim N(0,Q)$, $Z\sim N(0,N)$ and $\alpha$ is a constant number.Here we use $N$ to denote Gaussian distribution. Can anyone hint me?

Answer 1

Since $(X+S+Z, X+\alpha S)$ are jointly normal you can compute the entropy of a multivariate normal once you know its covariance matrix.

Answer 2

Thank qeschaton who gives me the wikipedia link where I find the answer. I list the conclusion below. \begin{aligned}h\left(f\right)&=-\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f(\mathbf {x} )\ln f(\mathbf {x} )\,d\mathbf {x} ,\\&={\frac {1}{2}}\ln \left(\left|2\pi e{\boldsymbol {\Sigma }}\right|\right)={\frac {k}{2}}\ln \left(2\pi e\right)+{\frac {1}{2}}\ln \left(\left|{\boldsymbol {\Sigma }}\right|\right)={\frac {k}{2}}+{\frac {k}{2}}\ln \left(2\pi \right)+{\frac {1}{2}}\ln \left(\left|{\boldsymbol {\Sigma }}\right|\right)\\\end{aligned}