normal random variable is a continuous random variable that allows you to manually specify the expectation and variance

constituents

• \(\mu\) the mean
• \(\sigma\) the variance

requirements

\begin{equation} X \sim \mathcal{N}(\mu, \sigma^{2}) \end{equation}

PDF:

\begin{equation} f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} \end{equation}

additional information

normal maximizes entropy

no other random variable uses as little parameters to convey as much information

Use a series of parametrized differentiable + invertible functions to transform simple distributions to complex ones.

Foreword

Hi there, internet traveler.

The time is 2015/2016, I was either in 5th or 6th grade. At that time, I was barely beginning to be actually comfortable using the language of English.

One of the ways I practiced English, which is also a habit I continue to do today, is to write. I write mostly expository prose now, but, back then, shining with childish naïvete, I decided to write a multi-part story as a means of practicing English.

\(\text{NP} \cap \text{coNP}: \forall x \in \qty {0,1}^{*}, \exists\) short, efficiently checkable proof of BOTH \(x\) presence/absence in \(L\)

some examples

• in P: PERFECT-MATCHING
• in P: PRIMES
• we don’t know if this is in \(P\): FACTORING … if it was, much of cryptography will break

A language \(B\) is NP-Complete if \(B \in NP\) and we have that every \(A \in NP\) has \(A \leq_{P} B\) with a polynomial time mapping reduction. We say \(B\) is NP-Hard if the reduction exists, and NP-Complete if \(B \in NP\) too.

Suppose a language \(L\) is NP-Complete, we have that every other language in \(NP\) is mapping reducable to \(L\). So, if \(L \in P\), then \(P= NP\), if \(L \not\in P\), then \(P \neq NP\).