Expected Value of Continuous Random Variables

Learning Outcomes

• Expectations

• Variance

• Properties

Continuous Random Variables

Expected Value

The expected value for a continuous distribution is defined as

\[ E(X)=\int x f(x)dx \]

The expectation of a function \(g(X)\) is defined as

\[ E\{g(X)\}=\int g(x)f(x)dx \]

Expected Value Properties

1. \(E(c)=c\), where \(c\) is constant
2. \(E\{cg(X)\}=cE\{g(X)\}\)
3. \(E\{g_1(X)+g_2(X)+\cdots+g_n(X)\}=E\{g_1(X)\}+E\{g_2(X)\}+\cdots+E\{g_n(X)\}\)

Variance

The variance of continuous variable is defined as

\[ Var(X) = E[\{X-E(X)\}^2] = \int \{X-E(X)\}^2 f(x)dx \]

Variance Properties

\(Y=aX+b\)

\(Var(Y) = Var(aX+b) = Var(aX) + Var(b) = a^2Var(X)\)

Uniform Distribution

Expected Value

\(X\sim\mathrm{U(a,b)}\)

\(a<x<b\)

\(f_X(x) = \frac{1}{b-a}\)

Variance

\(X\sim\mathrm{U(a,b)}\)

\(a<x<b\)

\(f_X(x) = \frac{1}{b-a}\)

Normal Distribution

Expected Value

\(X\sim\mathrm{N}(\mu, \sigma^2)\)

\(-\infty < x < \infty\)

\(f_X(x) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}\)

Variance

\(X\sim\mathrm{N}(\mu, \sigma^2)\)

\(-\infty < x < \infty\)

\(f_X(x) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}\)

Beta Distribution

Expected Value

\(X\sim\mathrm{Beta}(\alpha, \beta)\)

\(0<x<1\)

\(f_X(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}\)

Variance

\(X\sim\mathrm{Beta}(\alpha, \beta)\)

\(0<x<1\)

\(f_X(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}\)

\(\chi^2\)-Distribution

Expected Value

\(X\sim\chi^2_k\)

\(x>0\)

\(f_X(x)=\frac{x^{k/2-1}\exp\{-x/2\}}{2^{k/2}\Gamma(k/2)}\)

Variance

\(X\sim\chi^2_k\)

\(x>0\)

\(f_X(x)=\frac{x^{k/2-1}\exp\{-x/2\}}{2^{k/2}\Gamma(k/2)}\)