Understanding Uniform (Rectangular) Distribution

Uniform or Rectangular Distribution

Definition

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

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

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 $X \sim U(a, b)$.

Insight:

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

Graphical Representation

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

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

Mean and Variance

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

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

Insight:

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

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

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]$ 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: