STA256 Lecture 07

• Test Review:
  • Discrete Random Variable
    • Definition
    • Properties
  • Probability Mass Function
    • Closed form
    • Table
    • Graph
  • Cumulative Distribution Function
    • Closed form, piecewise
    • Graph looks like a staircase
  • Transformations - Univariate
• Example:
  • $P_X(x)=(\frac{1}{2}{)}^x$
  • Find the Probability Mass Function of $y=e^x\to g(x)=e^x$
  • $x=\ln (1)\to g^{-1}(x)=\ln (x)$
  • This is a one-to-one function
  • In stats, $\log ={\log }_e=\ln$
  • By the transfomation theorem, we have $P_Y(y)=P_X(g^{-1}(y))$
  • $P_X(x)={\left(\frac{1}{2}\right)}^x$
  • $g^{-1}(y)=\ln (y)$
  • $P_Y(y)={\left(\frac{1}{2}\right)}^{\ln (y)}$
  • Supported values is $y=e,e^2,e^3,\dots$, $y=\{x\in \mathbb{N}:e^x\}$
• Example:
  • Not one to one function #tk, options trading, what's the chance you blow the account?
  • Let $X$ have a Probability Mass Function
  • $\begin{array}{ccccc}X=x& -20& -10& 10& 20\\ P(X=x)& 0.10& 0.20& 0.4& 0.3\end{array}$
  • Find the Probability Mass Function of $Y$
  • $\underset{\text{The function }g}{\underbrace{Y=\left(\begin{array}{cc}-1& x<0\\ 1& x\ge 0\end{array}\right)}}$ not one-to-one
    • #tk why did we choose $y=\pm 1$?
  • By the transformation theorem
  • $P_Y(y)=P_X(g^{-1}(y))$
  • Support of $y=\pm 1$
  • $P_Y(-1)=P_X(g^{-1}(-1))=P_X(x=-10)+P_X(x=-20)$
  • $P_Y(-1)=P_X(g^{-1}(-1))=0.20+0.10$
  • $P_Y(-1)=P_X(g^{-1}(-1))=0.3$
  • $P_Y(1)=P_X(g^{-1}(1))=P_X(x=10)+P_X(x=20)$
  • $P_Y(1)=P_X(g^{-1}(1))=0.4+0.3$
  • $P_Y(1)=P_X(g^{-1}(1))=0.7$
  • The Probability Mass Function of $Y$
  • $\begin{array}{ccc}Y=y& -1& 1\\ P(Y=y)& 0.3& 0.7\end{array}$
• Continuous Random Variable
• Probability Density Function
• Percentile
• Quantile