expectation

consider a discrete random variable $\text{[math]}$ that receives the values $\text{[math]}$ with the probabilities $\text{[math]}$ respectively, the expected value of $\text{[math]}$ , denoted by $\text{[math]}$ , is defined as $\text{[math]}$ such that the series is absolutely convergent a more general formula: $\text{[math]}$ if $\text{[math]}$ is a constant random variable, i.e. it receives only one value $\text{[math]}$ with a probability of 1 then $\text{[math]}$ . if $\text{[math]}$ is constant, then for each random variable whose expected value is finite $\text{[math]}$ .

let $\text{[math]}$ and $\text{[math]}$ be two random variables defined over the same probability space, then:
$\text{[math]}$

let $\text{[math]}$ , we look at $\text{[math]}$ as the number of successes from a sequence of $\text{[math]}$ independent trials

different definitions that mostly say the same thing:

if a variable quantity $\text{[math]}$ can take on the particular values $\text{[math]}$ in $\text{[math]}$ mutually exclusive and exhausive situations with the respective probabilities $\text{[math]}$ to them, the the quantity
$\text{[math]}$ is called the expectation of $\text{[math]}$ .
[cite:@jaynes_prob_2003]

let $\text{[math]}$ be a real-valued random variable, if $\text{[math]}$ , then $\text{[math]}$ is called integrable and we call
$\text{[math]}$ the expectation or mean of $\text{[math]}$ . if $\text{[math]}$ , then $\text{[math]}$ is called centered.
[cite:@klenke_prob_2020]

the expected value of a random variable $\text{[math]}$ , denoted by $\text{[math]}$ is
$\text{[math]}$ provided that the integral or sum exists. if $\text{[math]}$ , we say that $\text{[math]}$ does not exist.
[cite:@berger_inference_2002]

if $\text{[math]}$ are constants and $\text{[math]}$ is a random variable, $\text{[math]}$ , more generally:
$\text{[math]}$