# How to calculate this normal distribution [duplicate]

I have this mass of probability is proportional to $$exp[-\frac{\beta}{2}\parallel x-Hz\parallel^{2}-\frac{\alpha}{2}z^{t}C^{t}Cz]$$ and I have to calculate to its distribution but I don't know how, its distribution is $$\mathcal{N}((\alpha C^{t}C+\beta H^{t}H)^{-1}\beta H^{t} x,(\alpha C^{t}C+\beta H^{t}H)^{-1}.$$ $z$ and $x$ are vectors and $H$ and $C$ matrices, Any idea?

• should the $||x-Hz||$ be $||x-Hz||^2$? And are you familiar with completing the square for vectors?
– jld
Dec 6 '17 at 22:17
• Could you explain your notation a bit more. what is $z$ for instance?
– Josh
Dec 6 '17 at 22:24
• For sure is squared but I don't know how to complete square for vectors. And x and z are a vectors, C and H are matrixs. Dec 6 '17 at 22:30
• matrices, not matrixs Dec 6 '17 at 23:02
• Do you know matrix algebra? You should expand $\Vert x-Hz \Vert^2$ as$$(x-Hz)^\text{T}(x-Hz)$$ Dec 7 '17 at 15:26