Intuitive meaning of $E(X^2)$ and $E(X+a)$? 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.

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*$E(X+a):$ 


is a mean value of some new random variable, exactly the $X+a$, which has just shifted ALL values of $X$ by $a$. Why is it happening? Just remember what is a formal definition of  some r.v. $X$, it's a function that takes values in $\Omega$ and map them into some subset of $\mathbb{R}$, i.e.: $$X:\Omega \to \mathbb{R}.$$
So we just move all that real values by $a$. And hence get a new r.v. with the mean $E(X+a)=E(X)+a.$


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*$E(X^2)$


Is similarly a mean of r.v. $X^2$. And how we get that random variable from $X$? We just square all of it ($X$) values, but the probability of each squared value will be the same as in $X$ without squaring. But in can be that $X$, for example, takes values $b$ and $-b$ with probability $p(b)$ and $p(-b)$, then in $X^2$ we will have $b^2$ and $b^2$ (which is the same), so the total probability of $b^2$ in $X^2$ will be $p(b)+p(-b).$
