1.8 Expectation of Random Variable

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Definition 1.8.1: Expectation

Continuous random variable

Let

• $X$ : Continuous random variable
• $f(x)$ : pdf of $X$
• ${\int }_{-\infty }^{\infty }|x|f(x)dx<\infty$

If $E(X)={\int }_{-\infty }^{\infty }xf(x)dx$

Then we say $E(X)$ is the expectation of $X$

Discrete random variable

Let

• $X$ : Discrete random variable
• $p(x)$ : pmf of $X$
• ${\sum }_x|x|p(x)<\infty$

If $E(X)={\sum }_{x}xp(x)$

Then we say $E(X)$ is the expectation of $X$

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Theorem 1.8.1: Expectation of a function

Let

• $X$ : Random variable

• $Y=g(X)$

Let $X$ is a Continuous random variable with pdf $f_X(x)$

If ${\int }_{-\infty }^{\infty }|g(x)|f_X(x)dx<\infty$

Then $E(Y)$ exists, given by $E(Y)={\int }_{-\infty }^{\infty }g(x)f_X(x)dx$

Let

• $X$ is a Discrete random variable, with
  • pmf $p_X(x)$
  • Support ${\mathcal{S}}_X$

If ${\sum }_{x\in {\mathcal{S}}_X}|g(x)|p_X(x)<\infty$

Then $E(Y)$ exists, given by $E(Y)={\sum }_{x\in {\mathcal{S}}_X}g(x)p_X(x)$

Theorem 1.8.2: Linearity of expectation

Let

• $X$ : Random Variable
• $g_1(X),g_2(X)$ : Functions of $X$

If $E(g_1(X)),E(g_2(X))$ exists

Then for any constants $k_1,k_2$, the following expectation exists

$$E[k_1{g}_1(X)+k_2{g}_2(X)]=k_{1}E[g_1(X)]+k_{2}E[g_2(X)]$$

This theorem proves that expectation is a linear operator. This allows us to easily do simple linear operations with expectations.