202405031126
Status: #idea
Tags: Statistics, Probability

Expected Value

Elementary Probability

It is a weighted average of all the values that a random variable under consideration can take. It is weighted to account for the it is computed as follows:

Where each probability is weighted by the actual value of its observation.

This value gives us on average, what would be the most likely value that $X$ would take.

In MAT3172

This definition makes use of what's called a Riemann-Stieltjes Integrals ~ Generalization of Riemann integrals where instead of differentiating directly with respect to $x$, we differentiate with respect to some non-decreasing function $\alpha$. Here we know by definition that a Cumulative Density Function ($F(x$)) is always non-decreasing, therefore we can integrate with respect to it.

Observe that the general form is the first equation, this form can be used for ANY cdf you toss at me, as in the worst case (if the cdf is not continuous) I can solve it by doing integration by parts, the latter two which are the equations we are typically given in #Elementary Probability for continuous random variables. They are used when we are dealing with a continuous random variable and therefore its cumulative density function is continuous, most of the time the functions we deal with in such classes will be both continuous and differentiable. But this does not have to be the case.

We can update the definition of variance in a similar fashion: Variance#In MAT3172

Interestingly, if $F(x)$ is non-negative, we can compute the expected value as:

Properties

1. $E(\sum c_i{X}_i)=\sum c_{i}E(X_i)$
2. $X\le Y\le Z⟹E(X)\le E(Y)\le E(Z)$

I spent an ungodly amount of time explaining expected value in Probability Measure (Based on Expected Value) so you can check the details there (at least until I bring it here.)

Relevant Links