What do differences in expectation value mean?

Question

I am working on some problems in Probability Theory but am a little bit confused with some of the concepts.

I understand that $E[X]$ is the expectation of a random variable. I also understand that to find the mean variance we need to square the difference from the expected value $E[X^2] - (E[X])^2$

What I do not understand is what it means when the values of this expectation are changed to look like:

$E[X^3]$ or $E[2X - 4]$ or $E[\sin(\pi X)]$

What does these mean and what effectively are we doing/changing?

Answer 1

Rather than taking the expectation of the random variable, you're now taking the expectation of the function of the random variable.

For instance, let's say $X$ were a fair 6-sided die. Then $$E[X^3] = \frac{1}{6}(1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3) = 73.5$$

We can interpret that as

The expected value I get for rolling a dice and taking the cubed value of that side of the dice is $73.5$.

The actual probability for each value depends on the probability density function $\mu$. To compute these expectations, you can set up your integral or summation much the same way, except instead of $x \cdot \mu(x)$ you put $f(x) \cdot \mu(x)$.