Title: Theory / Important / Math / Probability / Statistics

Name: The heaviside Step Function
Definition: Send \(x<0\) to \(0\)
\(0\) to \(\frac12\) and
\(x>0\) to 1
Page: http://mathworld.wolfram.com/HeavisideStepFunction.html
Context: Stats
Notation: \(H\)

Page: https://en.wikipedia.org/wiki/Bernoulli_distribution
Name: Bernouilli Distribution
Mean: \(p\)
Variance: \(p(1-p)\)
Skewness: \(\frac{1-2p}{\sqrt{p(1-p)}}\)
Kurtosis: \(\frac{1-6p(1-p)}{p(1-p)}\)
Moment generating function: \(1-p+pe^t\)
Characteristic function: \(1-p+pe^{it}\)
Variables: \(0\le p\le1\)
Intuition: A Boolean random variable
Mode: 0 if \(p<.5\)
\(0,1\) if \(p=.5\)
\(1\) otherwise
Cumulative function: Send negative to 0,
\((0,1)\) to \(1-p\) and
\([1,\infty)\) to 1.
Median: 0 if \(p<.5\)
\(.5\) if \(p=.5\)
\(1\) otherwise
Moment generating function computation: By definition
Variance computation: \(p-p^2=p(1-p)\)
Etymology: Swiss Mathematician