Expected Value

In Probability Theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average.

Expected Value (Definition)

Let $X$ be a discrete Random Variable with p.m.f. $f(x)$, then the expected value $\mathbb{E}[X]$ is

$\mu =\mathbb{E}[X]=\sum _{x\in X}x\cdot f(x)$

For a continuous Random Variable, we have $\mathbb{E}(x)={\int }_{-\infty }^{\infty }x(f(x))dx$

Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable.

When we change the $x$ term, it becomes below (notice that $f(x)$ doesn’t change). see property 2 below $\mathbb{E}(x^2)={\int }_{-\infty }^{\infty }{x}^2(f(x))dx$

Properties

1. For any constant $\alpha$ and $\beta$, $E(\alpha X+\beta Y)=\alpha E(X)+\beta E(Y)$ (linearity)
2. $E(g(x))={\sum }_{x\in X}g(x)\cdot f_x(x)$, $f_x(x)$ is the p.m.f. of $X$
   • What this means for example, if you have a p.d.f f(x) where
     • f(0) = 0.2
     • f(1) = 0.5
     • f(2) = 0.3
   • To find $E(X^2)$, you have $0^2\cdot 0.2+1^2\cdot 0.5+2^2\cdot 0.3=1.7$ (notice how the p.d.f. doesn’t change)
   • Another example: $E(\ln X)={\sum }_{x\in X}\ln xf(x)$

Other Properties

1. $E(k)=k$; $Var(k)=0$, where $k$ is a constant
2. $E(aX+b)=aE(X)+b;Var(aX+b)=a^{2}Var(X)$
3. $E(X-\mu )=0$ for any r.v. $X$
4. $Var(X)=E[(X-E(X){)}^2]=E(X^2)-{\mu }^2$
   1. To get to this answer, expand the squared term to $E[X^2-2XE(X)+E(X{)}^2]$, you can treat $E(X)$ as a constant, and realize that $E(x)=\mu$, so it simplifies to $E(x^2)-{\mu }^2$
   2. I was still a little confused, the proof is here: https://www.probabilitycourse.com/chapter3/3_2_4_variance.php#:~:text=The%20variance%20of%20a%20random,%E2%88%92%CE%BCX)2%5D.
5. If $X$ and $Y$ are independent then $Var(X+Y)=Var(X)+Var(Y)$

Related

• Variance

Random

Wow, another Serendipity moment. Tying the Reinforcement Learning stuff (expected return) I was seeing to Competitive Programming, as well as the probability theory that I will be learning for my upcoming Statistics class.