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  • Special Distributions

Continuous Uniform Distribution

• Notation $X\sim U({\theta }_1,{\theta }_2)$ or $X\sim unif({\theta }_1,{\theta }_2)$
• Where ${\theta }_1$ is the lower bound and ${\theta }_2$ is the upper bound
• Maybe Jensen's Inequality and Markov's Inequality can apply to find ${\theta }_i$?
• Probability Density Function:
  • $f_X(x)=\frac{1}{{\theta }_2-{\theta }_1}$ where ${\theta }_1\le x\le {\theta }_2$
• Properties:
  • Mean: $\mu =E[X]=\frac{{\theta }_1+{\theta }_2}{2}$
  • Variance: ${\sigma }^2=\frac{({\theta }_2-{\theta }_1{)}^2}{12}$
  • $P(a\le x\le b)={\int }_a^{b}f(x)dx$
    • Since the area is a rectangle it's just $A_{◻}=b\cdot h$
• Example:
  • Slide 51:
  • Informed that flight time is uniform between $2$ hours ($120$ minutes) and $2$ hours and $20$ minutes ($140$ minutes).
  • $X\sim U(120,140)$

        left=-10; right=200;
        top=0.1; bottom=-0.05;
        ---
        x=120
        x=140
        y=1/20
    

  • Claim: flight time is $2$ hours $5$ mintutes ($125$ minutes)
  • What's the probability they arrive $5$ minutes late
    • $\le 125+5$
    • $\le 130$ minutes late

          left=-10; right=200;
          top=0.1; bottom=-0.05;
          ---
          x=120
          x=140
          120<=x<=130|0<=y<= 1/20
          y=1/20
      

    • We want this area: $P(X\le 130)=bh$
    • $P(X\le 130)=(130-120)\left(\frac{1}{20}\right)$
    • $P(X\le 130)=\frac{10}{20}=\frac{1}{2}$
  • Find the probability that $P(x>135|x>130)$
  • $P(x>135|x>130)=\frac{P(X>135\cap X>130)}{P(X>130)}$
  • $P(x>135|x>130)=\frac{P(X>135)}{P(X>130)}$
  • $\frac{P(X>135)}{P(X>130)}=\frac{(140-135)\left(\frac{1}{20}\right)}{(140-130)\left(\frac{1}{20}\right)}=\frac{5}{10}=\frac{1}{2}$
• Example:
  • Same as above
  • Arrival time is $X\sim U[120,140]$
  • Find the ${95}^{\text{th}}$ percentile of arrival times
  • Let $x_{95}$ represent the ${95}^{\text{th}}$ Percentile of arrival times
  • At $x_{95}$, $95\mathrm{\%}$ of values are below it.

    	left=100; right=150;
    	top=0.1; bottom=-0.05;
    	---
    	x=120
    	x=140
    	x=139|dashed
    	(139,0)|label:x_95
    	y=1/20
    

  • At $x_{95}$
  • $P(X\le x_{95})=0.95$
  • ${\int }_{-\mathrm{\infty }}^{x_{95}}f(x)\text{differential}x$
  • $\dots$
  • Since it's uniform, at $x_{95}$, we have $P(X\le x_{95})=0.95$
  • $140-120=20$
  • $20(0.95)=19$
  • $120+19=139$
  • $(\text{base})(\text{height})=0.95$
  • $(x_{95}-120)\left(\frac{1}{20}\right)=0.95$
  • $\frac{x_{95}-120}{20}=0.95$
  • $x_{95}-120=19$
  • $x_{95}=139$