If an outcome can be from sets \(A=m\) or \(B=n\) with no overlaps, where \(A \cap B = \emptyset\), then, the total number of outcomes are \(|A| + |B| = m+n\)

If there are overlap:

\begin{equation} N = |A|+|B| - |A \cap B| \end{equation}

\(n\) random random variables are IID if they are

1. independent
2. identically distributed (see below)

“identically distributed”

Consider \(n\) random variables:

• \(X_i\) all have the same PMF / PDF
• and therefore, all have the same expectation and variance

central limit theorem

when things are IID, you can use central limit theorem.

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.