Uniform Distribution – Study Notes

Lesson Goals

  • Calculate probabilities for a continuous uniform distribution.
  • Compute the mean (μ) and standard deviation (σ) of a uniform distribution.

Discrete vs. Continuous Variables

  • Discrete random variable
    • Probability assigned to each individual outcome.
    • Finite or countably-infinite set of possible values.
  • Continuous random variable
    • Infinitely many possible values on a numerical continuum.
    • Impossible to assign a positive probability to a single point; doing so across an infinite set would push total probability above 11.
    • Probability is instead assigned to intervals.

Uniform Distribution Basics

  • A uniform distribution spreads probability evenly across a specified interval [a,b][a,b].
  • Every sub-interval of equal length has the same probability.
  • For continuous cases we work with a probability density function (pdf), not a probability mass function (pmf).

Probability Density Function (pdf)

  • Functional form:

    f(x)=\begin{cases}
    \dfrac{1}{b-a}, & a \le x \le b \
    0, & \text{otherwise}
    \end{cases}
  • Key implications:
    • Constant height 1b−a\dfrac{1}{b-a} across [a,b][a,b].
    • Integrates to 11 over the entire real line—i.e. ∫−∞∞f(x) dx=1\int_{-\infty}^{\infty}f(x)\,dx=1.

Mean and Standard Deviation

  • Mean (expected value):
    μ=a+b2\mu = \dfrac{a+b}{2}
  • Standard deviation:
    σ=b−a12\sigma = \dfrac{b-a}{\sqrt{12}}
  • Interpretation:
    • μ\mu locates the exact center of the interval.
    • σ\sigma measures spread; grows linearly with interval width b−ab-a.

Visual Representation & Area Interpretation

  • Graph of f(x)f(x) is a rectangle stretching from aa to bb on the x-axis with height 1b−a\dfrac{1}{b-a}.
  • Probability of an interval [c,d]⊆[a,b][c,d] \subseteq [a,b]:
    • Area of rectangle segment.
    • Formula: P(c≤X≤d)=(d−c)×1b−a=d−cb−aP(c \le X \le d)= (d-c)\times\dfrac{1}{b-a}=\dfrac{d-c}{b-a}.
    • No complex geometry required; area of a rectangle suffices.

Practical & Conceptual Significance

  • Applications: modeling scenarios where every outcome within a range is equally likely (e.g., random start time in an hour, random point on a stick).
  • Ethical / philosophical note: Emphasizes limitations of probability assignment; continuous outcomes necessitate interval-based thinking.
  • Connection to earlier probability principles: Upholds total-area-equals-one rule and demonstrates transition from discrete pmf to continuous pdf.

Key Formulas Summary

  • pdf: f(x)=1b−a  for a≤x≤bf(x)=\frac{1}{b-a}\;\text{for }a\le x\le b
  • Mean: μ=a+b2\mu=\frac{a+b}{2}
  • Standard deviation: σ=b−a12\sigma=\frac{b-a}{\sqrt{12}}
  • Interval probability: P(c≤X≤d)=d−cb−aP(c \le X \le d)=\frac{d-c}{b-a}

Mini Check-Your-Understanding (Self-Test)

  • If a=2a=2 and b=8b=8, find:
    • P(3≤X≤5)P(3 \le X \le 5) → 5−38−2=26=13\dfrac{5-3}{8-2}=\dfrac{2}{6}=\dfrac{1}{3}.
    • μ\mu → 2+82=5\dfrac{2+8}{2}=5.
    • σ\sigma → 8−212=612=623=33=3\dfrac{8-2}{\sqrt{12}}=\dfrac{6}{\sqrt{12}}=\dfrac{6}{2\sqrt{3}}=\dfrac{3}{\sqrt{3}}=\sqrt{3}.

Bottom Line

  • Uniform distribution = equal likelihood across an interval.
  • Probabilities correspond to areas; means and SDs follow direct formulas.
  • Mastery of this distribution provides a stepping stone to understanding more complex continuous distributions.