I understand conceptually that $E(X)$ is the average expected value of random variable $X$ over multiple trials over a long period of time i.e. the mean. Similarly, I understand for conditionals that $E(X|Y)$ is the average value of $X$ for all cases where $Y$ has already happened.
However, what is the intuitive meaning behind expressions like $E(X^2)$, and other expectations besides $E(X)$? For instance, the meaning of $E(X+a)$, where $a$ is a constant? I can't seem to grasp the concept of those expressions.
Best Answer
$X^2$ or $X + a$ are both random variables, pretty much the same as $X$. In the first case, you are to compute the mean value of $X^2$ and in the second one, you should compute the mean of $X + a$.
A little example:
Let $X$ be a random variable that takes value $-2$ with probability $\dfrac{1}{2}$, $1$ with probability $\dfrac{1}{3}$ and $3$ with probability $\dfrac{1}{6}$. Hence, $$E(X) = -2\cdot \dfrac{1}{2} + 1 \cdot \dfrac{1}{3} + 3 \cdot \dfrac{1}{6} = -\dfrac{1}{6}$$
Then $X^2$ is $(-2)^2$ with probability $\dfrac{1}{2}$, $1^2$ with probability $\dfrac{1}{3}$ and $3^2$ with probability $\dfrac{1}{6}$.
$$E(X^2) = 4\cdot \dfrac{1}{2} + 1 \cdot \dfrac{1}{3} + 9 \cdot \dfrac{1}{6} = \dfrac{23}{6}$$
As regards to, say, $X+2$, we can merely write down $$E(x+2) = E(x) + 2 =\dfrac{11}{6}$$