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  • STA256 Lecture 09

Expectation of a Function of a random variable

• $X$ is a random variable.
• Let $Y=g(X)$ for some function $g$
• #tk slides 94 theorem 35
• Example:
  • Discrete Random Variable
    • Let $X$ have Probability Mass Function given by:
    • $\begin{array}{cccc}& x_1& x_2& x_3\\ X=x& 1& 2& 3\\ P(X=x)& \frac{1}{4}& \frac{1}{2}& \frac{1}{4}\\ & P(x_1)& P(x_2)& P(x_3)\end{array}$
    • Let $g(x)=2x^2+1$ find $E(g(x))$
    • $E(x)=\sum x_{i}P(x_i)$
    • $E(g(x))=\sum g(x_i)P(x_i)$
      • $g(x_1)P(x_1)+g(x_2)P(x_2)+g(x_3)P(x_3)$
      • $(2(1{)}^2+1)\left(\frac{1}{4}\right)+(2(2{)}^2+1)\left(\frac{1}{2}\right)+(2(3{)}^2+1)\left(\frac{1}{4}\right)$
      • $(3)\left(\frac{1}{4}\right)+(9)\left(\frac{1}{2}\right)+(19)\left(\frac{1}{4}\right)$
      • $10$
  • Continuous Random Variable
    • Let $X$ be a Continuous Random Variable with a Probability Density Function given by:
      • $f(x)=\left\{\begin{array}{cc}\frac{19}{4}-x^3& 1\le x\le 2\\ 0& \text{otherwise}\end{array}\right\}$
      • Let $g(x)=\frac{1}{x}$ find $E(g(x))$
      • $E(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx$
      • $E(g(x))={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(x)f(x)dx$
      • $E(g(x))={\int }_1^2\frac{1}{x}(\frac{19}{4}-x^3)dx$
      • $E(g(x))={\int }_1^2(\frac{19}{4x}-x^2)dx$
        • Antiderive
        • $(\frac{19}{4}\ln x+\frac{1}{3}{x}^3){}_1^2$
        • $(\frac{19}{4}\ln 2+\frac{1}{3}(2{)}^3)-(\frac{19}{4}\ln 1+\frac{1}{3}(1{)}^3)$
        • $(\frac{19}{4}\ln 2+\frac{1}{3}(2{)}^3)-(\frac{1}{3}(1{)}^3)$
        • $(\frac{19}{4}\ln 2+\frac{8}{3})-(\frac{1}{3})$
        • $\frac{19}{4}\ln 2+\frac{7}{3}$