Probability Theory

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

example:

coin flip

∑={e:H,T}→12\begin{gather*} \sum=\{e:H,T\}\rightarrow\frac{1}{2} \end{gather*}

Sum of all probability of your space should be 1

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

We have a random variable XX. You have an event in that random variable space called xx. The probability of finding that event in that random variable distribution

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

where

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

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

Note that the probabilities are always between 0 and 1

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