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  • STA256 Lecture 08

Expected Value

Univariate - Random Variables

• Similar to average of a population
• The mean
• Properties
  • It's linear
• Discrete Random Variable
  • $\begin{array}{cccccc}X=x& 0& 1& 2& 3& 4\\ P(X=x)& 0.1& 0.3& 0.3& 0.25& 0.05\\ & P(x_1)& P(x_2)& \dots & \dots & P(x_5)\end{array}$
  • Find the mean or Expected Value (mean the same thing)
  • $\mu =E(X)=\sum X_i\cdot P(x_i)$
  • $=(0)(0.1)+(1)(0.3)+\cdots +(4)(0.05)$
  • $=1.85$
  • On average the store will sell $1.85$ TVs per day.
• Continuous Random Variable
  • Let Continuous Random Variable $X$ have Probability Density Function
  • $f(x)=\left\{\begin{array}{cc}\frac{x}{4}-\frac{1}{2}& 3\le x\le 5\\ 0& \text{otherwise}\end{array}\right\}$
  • Find the mean
  • $\mu =E(X)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx$
  • $={\int }_3^{5}x(\frac{x}{4}-\frac{1}{2})dx$
  • $(\frac{1}{12}{x}^3-\frac{1}{4}{x}^2){}_3^5$
  • $\frac{25}{6}$

Bivariate - Random Vector

• $E(X_1,X_2)=\{\begin{array}{l}\sum _{\mathrm{\forall }{x}_2}\sum _{\mathrm{\forall }{x}_1}{x}_1{x}_{2}p(x_1,x_2)\\ \underset{\mathrm{\forall }{x}_2}{\int }\underset{\mathrm{\forall }{x}_1}{\int }{x}_1{x}_{2}f(x_1,x_2)d{x}_{1}d{x}_2\end{array}$
• Examples on slides 23-25