random variable

klenke's definition is rooted in measure theory.

let $\text{[math]}$ be a measurable space and let $\text{[math]}$ be measurable.

$\text{[math]}$ is called a random variable with values in $\text{[math]}$ . if $\text{[math]}$ , then $\text{[math]}$ is called a real random variable or simply a random variable.
for $\text{[math]}$ , we denote $\text{[math]}$ and $\text{[math]}$ . in particular, we let $\text{[math]}$ and define $\text{[math]}$ similarly and so on.

[cite:@klenke_prob_2020 definition 1.102]

some stuff from college1

a random variable X is a function $\text{[math]}$ from the sample space to the real line ( $\text{[math]}$ ), such that every event in the sample space corresponds to a real number. $\text{[math]}$ is the probability of receiving the value $\text{[math]}$ and $\text{[math]}$ is the probability function

there are two types of random variables

we roll a symmetric coin 3 times
$\text{[math]}$ we define $\text{[math]}$ as the number of times that "heads" was received
$\text{[math]}$ is a random variable with 4 possible values, $\text{[math]}$ , $\text{[math]}$

let $\text{[math]}$ be the frequency of the value $\text{[math]}$ , we get $\text{[math]}$ , so $\text{[math]}$
let $\text{[math]}$ be the probability to get the value $\text{[math]}$ , so $\text{[math]}$ , we get $\text{[math]}$
$\text{[math]}$

let $\text{[math]}$ be arbitrary values and $\text{[math]}$ their frequencies
the variances of $\text{[math]}$ is the measure of their scattering around the average
$\text{[math]}$

we build off from the previous lemmas:
$\text{[math]}$ so
$\text{[math]}$

$\text{[math]}$ is the expected value of a random variable $\text{[math]}$ , which is the average of it, so we get $\text{[math]}$ so $\text{[math]}$

on his way to work a person goes through 3 traffic lights, the probability of the first being green is 0.6, the second 0.5, the third 0.4
let $\text{[math]}$ be the number of traffic lights they pass without waiting (green)

find the probability function of the random variable $\text{[math]}$

$\text{[math]}$ can equal one of $\text{[math]}$
$\text{[math]}$ (no green)
$\text{[math]}$ (1 green)
$\text{[math]}$ (2 green)
$\text{[math]}$ (3 green)
so $\text{[math]}$

find the expected value and the variance of $\text{[math]}$

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$