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Uniform Distribution (Continuous)

Definition / Introduction

  • The Continuous Uniform distribution describes a Continuous Random Variable where all possible values within a specified range \([a, b]\) are equally likely.
  • It represents complete uncertainty within the bounds \(a\) and \(b\); no single value or sub-interval within the range is favored over another of the same size.
  • Think of it as randomly picking a real number from a fixed interval.

Key Concepts

1. The Uniform Random Variable

  • A continuous random variable \(X\) follows a Uniform distribution on the interval \([a, b]\) if its probability is spread evenly across that interval.
  • Notation: \(X \sim \text{Uniform}(a, b)\) or \(X \sim U(a, b)\).

2. Probability Density Function (PDF)

  • The PDF is constant within the interval \([a, b]\) and zero elsewhere. The height of the constant value must make the total area under the curve equal to 1. $$ f(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \le x \le b \ 0 & \text{otherwise} \end{cases} $$
  • The width of the interval is \((b-a)\). The height \(\frac{1}{b-a}\) ensures the area (width \(\times\) height) is \((b-a) \times \frac{1}{b-a} = 1\).

3. Parameters

  • The Uniform distribution has two parameters:
    • \(a\): The minimum possible value (lower bound).
    • \(b\): The maximum possible value (upper bound), where \(b > a\).

4. Cumulative Distribution Function (CDF)

  • The CDF \(F(x) = P(X \le x)\) represents the accumulated probability up to \(x\). $$ F(x) = \begin{cases} 0 & \text{if } x < a \ \frac{x-a}{b-a} & \text{if } a \le x \le b \ 1 & \text{if } x > b \end{cases} $$
  • Intuition: The CDF increases linearly from 0 at \(x=a\) to 1 at \(x=b\). The proportion of the interval covered up to \(x\) is \(\frac{x-a}{b-a}\).

5. Expected Value (Mean)

  • The expected value (average) is the midpoint of the interval: $$ E[X] = \frac{a+b}{2} $$

6. Variance

  • The variance (measure of spread) is: $$ Var(X) = \frac{(b-a)^2}{12} $$
  • The variance depends only on the width of the interval \((b-a)\).

7. Calculating Probabilities

  • The probability of \(X\) falling within a sub-interval \([c, d]\) (where \(a \le c \le d \le b\)) is the area under the PDF from \(c\) to \(d\): $$ P(c \le X \le d) = \int_c^d \frac{1}{b-a} \, dx = \frac{d-c}{b-a} $$
  • This is simply the ratio of the length of the sub-interval \((d-c)\) to the length of the total interval \((b-a)\).

Connections to Other Topics

  • The Standard Uniform Distribution is a special case where \(a=0\) and \(b=1\), denoted \(U(0, 1)\). This is fundamental in computer science for generating random numbers. Pseudo-random number generators often produce values that approximate a \(U(0, 1)\) distribution.
  • Random variables from any continuous distribution can be generated using a \(U(0, 1)\) generator and the Inverse Transform Sampling method based on the target distribution's CDF. If \(U \sim U(0,1)\), then \(X = F^{-1}(U)\) has CDF \(F\).
  • Used as a non-informative prior distribution in Bayesian Statistics in some contexts.
  • The [[11_Beta_Distribution|Beta distribution]] with \(\alpha=1, \beta=1\) reduces to \(U(0,1)\).

Summary

  • Models outcomes that are equally likely within a fixed range \([a, b]\).
  • Parameters: \(a\) (min value), \(b\) (max value).
  • PDF: Constant \(\frac{1}{b-a}\) between \(a\) and \(b\), zero otherwise (rectangular shape). Uses \begin{cases}.
  • Mean: \(\frac{a+b}{2}\) (midpoint).
  • Variance: \(\frac{(b-a)^2}{12}\).
  • Probability over a sub-interval \([c, d]\) is \(\frac{d-c}{b-a}\).
  • Crucial for random number generation (\(U(0, 1)\)).

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