1. Probability and Random Variables

Sample space

The sample space, $S/\Omega$, of the experiment is the set of all possible outcomes $$ S/\Omega = \{ s_1, s_2, ..., s_n \} $$

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Event

An event is a subset of the [[Sample space|sample space]], or otherwise a set of possible outcomes.

$$P(A)=\sum _{\{\omega \in A\}}P(\omega )$$ Link to original

Probability Measure

A probability measure on $S$ is a function $\text{Prob: }S\to [0,1]$ such that the sum of the values of the output of the function for each outcome is $1$; $\text{Prob}(s_1)+\text{Prob}(s_2)+\dots +\text{Prob}(s_n)=1$.

We can extend the probability measure to events: $\text{Prob: }\mathcal{P}(S)\to [0,1]$ such that the sum is once again $1$; $\text{Prob}(E)={\sum }_{s\in E}\text{Prob}(s)$.

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Random Variable

A random variable on $S$ is a measurement of a random outcome $s\in S$.

The probability that a random variable $X$ takes the value $k\in \mathbb{R}$ is given by:

$$\text{Prob}(X=k)=\text{Prob}(\{s\in S:X(s)=k\})$$

It follows similarly that:

$$\begin{array}{c}\text{Prob}(X\le k)=\text{Prob}(\{s\in S:X(s)\le k\})\\ \text{Prob}(X\ge k)=\text{Prob}(\{s\in S:X(s)\ge k\})\end{array}$$Link to original

Probability Mass Function

The probability mass function of a random variable $X$ is a function which says how likely a given value is to appear as the measurement of a random event. $p_X(k)=\text{Prob}(X=k)$.

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Expectation / Variance

Expectation of Variable

The expectation of a random variable $X$ is the weighted average of the possible values of $X$

$$E[X]=\sum _{k}k\cdot \text{Prob}(X=k)$$

Given $X$ and $Y$ are any two random variables and $a,b\in \mathbb{R}$, it follows:

$$\begin{array}{rl}E[aX+b]& =aE[X]+b\\ E[X+Y]& =E[X]+E[Y]\end{array}$$Link to original

Markov's Inequality

Markov’s Inequality: given $X$ is a random variable with expectation $E[X]$:

$$\text{Prob}(X\ge a\cdot E[X])\le \frac{1}{a}\text{ for any }a>0$$Link to original

Variance

The variance of a random variable $X$ is a measure of the expected deviation from the average $\mu =E[X]$.

$$\begin{array}{rl}Var[X]& =E[(X-\mu {)}^2]\\ & =E[X^2]-(E[X]{)}^2\end{array}$$Link to original

Standard Deviation

We can therefore, find the standard deviation of a random variable $X$ by taking the square-root of the Variance of $X$:

$${\sigma }_x=\sqrt{Var[X]}$$Link to original