Understanding Uniform (Rectangular) Distribution

Uniform or Rectangular Distribution

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Definition

A continuous random variable XX is said to follow a rectangular (uniform) distribution if its probability density function (pdf) is given by:

f(x)=1baif axbf(x) = \frac{1}{b-a} \quad \text{if } a \leq x \leq b

Insight:

  • This function describes a constant probability over the interval $$$a, b]$.
  • Outside of this range, the probability is zero.

Notation

It is symbolically written as XU(a,b)X \sim U(a, b).

Insight:

  • The notation U(a,b)U(a, b) signifies a uniform distribution over the interval \[a,b]\[a, b].

Graphical Representation

The graph of f(x)f(x) is a rectangle:

  • Height: 1ba\frac{1}{b-a}
  • Base: axba \leq x \leq b

Mean and Variance

For a uniform distribution XU(a,b)X \sim U(a, b):

Mean (Expected Value)=a+b2\text{Mean (Expected Value)} = \frac{a + b}{2}

Insight:

  • The mean is the midpoint of the interval \[a,b]\[a, b].

Variance=(ba)212\text{Variance} = \frac{(b-a)^2}{12}

Insight:

  • The variance measures the spread of the distribution.
  • It is derived from the average of squared deviations from the mean.

Additional Information

  • Applications: This distribution is commonly used in simulations and modeling scenarios where each outcome in the interval \[a,b]\[a, b] is equally likely.
  • Properties: It is symmetric around the mean, has finite variance, and is easy to integrate due to its constant density function over the interval.

Conclusion: Understanding uniform distribution is key for analyzing scenarios where outcomes are equally probable over a specified range.

Extended readings:

en.wikipedia.org
Continuous uniform distribution - Wikipedia
stats.libretexts.org
4.3: Uniform Distributions - Statistics LibreTexts
www.ucd.ie
[PDF$$ Statistics: Uniform Distribution (Continuous)