Uniform Distribution (Continuous)

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The Continuous Uniform distribution is the simplest of all continuous distributions. It models a scenario where every single possible decimal value within a specific range has the exact same probability density.

Core Concepts

If a random variable $X$ follows a Continuous Uniform distribution, we write:

X \sim \mathcal{U}(a, b)

Where:

$a$ is the minimum possible value.
$b$ is the maximum possible value.

Probability Density Function (PDF)

Because every value is equally likely, the PDF is a completely flat horizontal line between $a$ and $b$ . Because the total area under the line must equal exactly $1$ , the height of the line is simply $1$ divided by the width of the interval.

f(x) = \begin{cases} \frac{1}{b - a} & \text{for } a \le x \le b \\ 0 & \text{for } x < a \text{ or } x > b \end{cases}

Cumulative Distribution Function (CDF)

The CDF grows perfectly linearly (like a straight ramp) from $0$ to $1$ between $a$ and $b$ .

F(x) = \begin{cases} 0 & \text{for } x < a \\ \frac{x - a}{b - a} & \text{for } a \le x \le b \\ 1 & \text{for } x > b \end{cases}

Key Metrics

Expected Value (Mean): $E[X] = \frac{a + b}{2}$ (Exactly in the middle)
Variance: $\text{Var}(X) = \frac{(b - a)^2}{12}$

Real-World Examples

Random Number Generators: The foundational Math.random() function in programming languages generates a continuous uniform distribution between $0$ and $1$ .
Waiting for a Train: If a train comes exactly every 20 minutes, and you arrive at the station at a completely random time, your wait time is uniformly distributed between $0$ and $20$ minutes.

Interactive Visualization

Use the sliders below to adjust the minimum ( $a$ ) and maximum ( $b$ ) boundaries. Notice how the total width changes, and in response, the height (density) perfectly scales up or down to ensure the total area of the pink rectangle remains exactly $1.0$ .

Continuous Uniform Distribution

A flat density on [a, b] (all values equally likely).

Minimum (a)

Maximum (b)

Test Your Knowledge

Example: Continuous Waiting Time

A subway train arrives precisely every 15 minutes. If you arrive at the station at a totally random time, your waiting time $X$ is uniformly distributed between 0 and 15. What is the probability you have to wait more than 10 minutes?

View Step-by-Step Solution

This is a Continuous Uniform Distribution $U(0, 15)$ .

The PDF is $f(x) = \frac{1}{b-a} = \frac{1}{15-0} = \frac{1}{15}$

We want $P(X > 10)$ , which is the area under the curve from 10 to 15: $\text{Base} \times \text{Height} = (15 - 10) \times \frac{1}{15} = 5 \times \frac{1}{15} = \frac{5}{15} = \frac{1}{3}$

There is a 33.3% chance you will wait more than 10 minutes.

Discrete Uniform Normal Distribution