Expected Value

Topics: Probability

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(definition)

Let $X$ be a real random variable defined on $(\Omega ,\mathcal{F},\mathbb{P})$.

The expected value of a random variable $X$, denoted $\mathbb{E}[X]$, is defined as follows.

The expected value has several properties.

Discrete Case §

When $X$ is a discrete random variable, we say that the expected value of $X$, $\mathbb{E}[X]$, exists if:

$$\sum _{x\in R_X}|x|f_X(x)<\infty$$

In such a case:

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

Continuous Case §

When $X$ is a continuous random variable, we say that the expected value of $X$, $\mathbb{E}[X]$, exists if:

$${\int }_{\mathbb{R}}|x|f_X(x)dx<\infty$$

In such a case:

$$\mathbb{E}[X]={\int }_{\mathbb{R}}x{f}_X(x)dx$$