Properties of the Expected Value

Topics: Expected Value - Probability

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(theorem)

Let $X$ be a random variable and $a,b\in \mathbb{R}$. Then, the expected value of $X$ has the following properties

1. If $\mathbb{P}(X\ge 0)=1$ ($X$ always positive), then $\mathbb{E}[X]\ge 0$.
2. If $X=a$ ($X$ is constant), then $\mathbb{E}[X]=a$
3. $\mathbb{E}[aX+b]=a\mathbb{E}[X]+b$
4. If $\mathbb{E}[X]$ exists, then $|\mathbb{E}[X]|\le \mathbb{E}[|X|]$
5. If $g$ and $h$ are functions such that $g(x)$ and $h(x)$ are random variables whose values exist, if $g(x)\le h(x)\forall x$, then $\mathbb{E}[g(x)]\le \mathbb{E}[h(x)]$