klenke's definition is rooted in measure theory.
let be a measurable space and let be measurable.
- is called a random variable with values in . if , then is called a real random variable or simply a random variable.
- for , we denote and . in particular, we let and define similarly and so on.
some stuff from college1
a random variable X is a function from the sample space to the real line ( ), such that every event in the sample space corresponds to a real number. is the probability of receiving the value and is the probability function
there are two types of random variables
- discrete random variable
- continuous random variable
we roll a symmetric coin 3 times we define as the number of times that "heads" was received is a random variable with 4 possible values, ,
let be the frequency of the value , we get , so let be the probability to get the value , so , we get
let be arbitrary values and their frequencies the variances of is the measure of their scattering around the average
we build off from the previous lemmas:
so
is the expected value of a random variable , which is the average of it, so we get so
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 be the number of traffic lights they pass without waiting (green)
find the probability function of the random variable
can equal one of (no green) (1 green) (2 green) (3 green) so
find the expected value and the variance of