Here’s a list of all indexes:

• Projects Index
• Research Index
• Production Index
• About

This should be reflected on a fancier way on my home page.

voltage across a inductor

\begin{equation} V = \epsilon = -L \dv{I}{t} \end{equation}

this is kind of a formulation of faraday’s law.

\begin{equation} I(t) = \frac{V_0}{R_1} (1-e^{\frac{-t}{\frac{L}{R}}}) \end{equation}

energy stored in an inductor

\begin{equation} E = \frac{1}{2} LI^{2} \end{equation}

inference is the act of updating the distribution of a random variable based on distribution of actually observed variables:

\begin{equation} P(X|Y) \end{equation}

where \(Y\) is observed, and we want to know how likely \(X\) would therefore be.

We call the set \(X\) the “query variables”, \(Y\) as “evidence varibales”, and anything that we didn’t use which connects the two variables as “hidden variables”.

If things are not in the right order of \(X\) and \(Y\), consider the Bayes rule.

If we know that \(a,b\) are both Gaussian distributions, then we have that:

\begin{equation} \mqty[a \\ b] \sim \mathcal{N} \qty(\mqty[\mu_{a} \\mu_{b}], \mqty[A & C \\ C^{T} & B]) \end{equation}

whereby:

• \(A\) is the covariance of each element of \(A\)
• \(B\) is the covariance of each element of \(B\)
• \(C\) is the covariance of \(A\) against \(B\)

To perform inference:

\begin{equation} p(a|b) = \mathcal{N}(a | \mu_{a|B}, \Sigma_{a|b}) \end{equation}

wherby:

\begin{equation} \mu_{a|b} = \mu_{a} + CB^{-1}(b-\mu_{b}) \end{equation}