Probability Theory

probability space $\Sigma$ has points and elements and associated probabilities with them

example:

coin flip

\begin{gather*} \sum=\{e:H,T\}\rightarrow\frac{1}{2} \end{gather*}

Sum of all probability of your space should be 1

\begin{gather*} \sum P(\Sigma)=1 \end{gather*}

We have a random variable $X$ . You have an event in that random variable space called $x$ . The probability of finding that event in that random variable distribution

\begin{gather*} P(X)=\sum_{e\in V_x}P(e) \end{gather*}

where

\begin{gather*} V_x=\{e:X(e)=x\} \end{gather*}

Some famous probability distributions (that are used in this guide) are

Poisson Distribution
Gaussian Distribution

Note that the probabilities are always between 0 and 1

\begin{gather*} 0\leq P(x)\leq 1 \end{gather*}