How to calculate $E[X^2]$ for a die roll? Apparently:
$$
E[X^2] = 1^2 \cdot \frac{1}{6} + 2^2 \cdot \frac{1}{6} + 3^2\cdot\frac{1}{6}+4^2\cdot\frac{1}{6}+5^2\cdot\frac{1}{6}+6^2\cdot\frac{1}{6}
$$
where $X$ is the result of a die roll.
How come this expansion?

There are various ways to justify it.
For example, it follows from the definition of expectation and the law of the unconscious statistician.
Or consider the case $Y=X^2$ and computing $E(Y)$.
