STA256 Lecture 06

STA256H5F_Fall2025_Ch1_Probability_Distributions.pdf
Textbook 1.6

• Review
  • Random Variables
    • Discrete Random Variable
    • Continuous Random Variable
  • Probability Distribution Function
    • Probability Mass Function
    • Cumulative Distribution Function
• Example:
  • Cumulative Distribution Function
  • $\begin{array}{cccccc}X=x& 0& 1& 2& 3& 4\\ P(X=x)& 0.1& 0.3& 0.3& 0.25& 0.05\end{array}$
  • Find and graph the cdf of the distribution
  • $F(0)=P(X\le 0)$
  • $F(0)=0.1$
  • $F(1)=0.4$
  • $F(2)=0.7$
  • $F(3)=0.95$
  • $F(4)=1$
  • Closed form:
    • If $x<0$, then we get the probability of nothing, no data in that range.
    • $F(x)=\left\{\begin{array}{cc}0& x<0\\ 0.1& 0\le x<1\\ 0.4& 1\le x<2\\ 0.7& 2\le x<3\\ 0.95& 3\le x<4\\ 1& x\ge 4\end{array}\right\}$
  • Sketch:
  • Find $F(2)=0.7$ considering the piecewise function
  • Find $F(2.73)=0.7$
  • Use CDF to find the probability $P(X>3)$
    • $P(X>3)=1-P(X\le 3)$
    • $P(X>3)=1-F(3)$
    • $P(X>3)=1-(0.95)$
    • $P(X>3)=0.05$
  • Use CDF to find the probability that $P(1<x\le 3)$
    • Since $x$ is discrete and CDF is right continuous
    • $P(1<x\le 3)=F(3)-F(1)$
    • $P(1<x\le 3)=0.95-0.4$
    • $P(1<x\le 3)=0.55$
• Transformations - Univariate