What is $E(X^2)$ mean in literal terms?

Question

In my probability and statistics class we learned about expected value, or $E(X)$. We also did some work about finding expected values of functions and such, like $E(g(x))$. And in the case of finding the variance, one of the steps involve finding $E(X^2)$. Does this mean anything in real-life terms?

Using a geometric distribution as an example, is $E(X^2)$ the expected number of times a trial needs to happen until the event $X$ happens twice in a row?

Answer 1

Here's an example. Suppose $X$ is a random variable that represents the outcome of a roll of a die numbered $1$ to $6$ inclusive. No assumption is made about the fairness of the die. Then $X^2$ is a random variable that represents the outcome of the square of the roll; whereas $$X \in \{1, 2, 3, 4, 5, 6\},$$ we have $$X^2 \in \{1, 4, 9, 16, 25, 36\}.$$

Now, if we suppose that this die is indeed fair, then we can easily compute the expected value of $X$: $$\operatorname{E}[X] = \sum_{x=1}^6 x \Pr[X = x] = \frac{1}{6} (1 + 2 + 3 + 4 + 5 + 6) = \frac{7}{2}.$$ The expectation of $X^2$ is: $$\operatorname{E}[X^2] = \sum_{x=1}^6 x^2 \Pr[X = x] = \frac{1}{6}(1 + 4 + 9 + 16 + 25 + 36) = \frac{91}{6}.$$ This should give you some intuition behind the meaning of $X$ versus $X^2$ and their corresponding expectations.

Answer 2

Using a geometric distribution as an example, is $E(X^2)$ the expected number of times a trial needs to happen until the event $X$ happens twice in a row?

No, it doesn't mean anything as convenient as that.   It is, in this case, the expectation of the square of the count of trials until the success.   We can name this the "second raw moment", but what the heck does that mean?

Now what does have intuitive meaning is some concept of deviation from the expected value as an indicative measure of "spread" or randomness.   Do you get the feeling for this: that the more the data values can stray from the mean (in either direction) the greater will be the variation in the data?   So how might we measure this concept?

Well, Variance, is defined as the expectation of the square of the distance to the value from the mean .   Is that sensible?

$$\begin{align}\mathsf{Var}(X) &= \mathsf E(~(X-\mathsf E(X))^2~)\end{align}$$

Expanding that out and applying the Linearity of Expectation we obtain the final result. $$\begin{align}\mathsf{Var}(X) &= \mathsf E(~(X-\mathsf E(X))^2~) \\ &= \mathsf E(X^2-2X\mathsf E(X)+\mathsf E(X)^2) \\ &= \mathsf E(X^2)-2\mathsf E(X)\mathsf E(X)+\mathsf E(X)^2 \\ &= \mathsf E(X^2)-\mathsf E(X)^2\end{align}$$

That's how $\mathsf E(X^2)$ (called the "second raw moment") relates to variance (called the "second central moment").

Answer 3

Imagine that $X$ is the side length of a square. Then $X^2$ is its area. Now $E(X)$ is the expected side length and $E(X^2)$ its expected area. It turns out the square of the expected length is not the expected area. The difference is called variance.