Exercise 4.6 - Bivariate Gaussian

Answers

We have:

\begin{aligned} p (x_{1}, x_{2}) & = \frac{1}{2 π \sqrt{| Σ |}} \cdot \exp (- \frac{1}{2} {(𝐱 - μ)}^{T} Σ (𝐱 - μ)) \\ = \frac{1}{2 π σ_{1} σ_{2} \sqrt{1 - ρ^{2}}} \cdot \exp (- \frac{1}{2} (\begin{matrix} x_{1} - μ_{1} & x_{2} - μ_{2} \end{matrix}) (\begin{matrix} σ_{1}^{2} & ρ σ_{1} σ_{2} \\ ρ σ_{1} σ_{2} & σ_{2}^{2} \end{matrix}) (\begin{matrix} x_{1} - μ_{1} \\ x_{2} - μ_{2} \end{matrix})) . \end{aligned}

solour_lfq

2021-03-24 13:42