The Uniform or Rectangular Distribution

STA 211 · Lecture 11

Introduction

A continuous uniform distribution assigns equal density to all points in a finite interval. It is appropriate when only the lower and upper bounds are known and no value within the interval is favoured.

Learning Objectives

You should be able to:

• state the uniform PDF and CDF;
• calculate interval probabilities;
• derive the mean and variance; and
• transform a uniform random variable linearly.

Density and Distribution Functions

If $X \sim U(a,b)$ with $a < b$, then

$$f_{X}(x) = \left\{ \begin{matrix} \frac{1}{b - a}, & a \leq x \leq b, \\ 0, & \text{otherwise}. \end{matrix} \right.\$$

Uniform probability density function

Uniform probability density function

The CDF is

$$F_{X}(x) = \left\{ \begin{matrix} 0, & x < a, \\ \frac{x - a}{b - a}, & a \leq x \leq b, \\ 1, & x > b. \end{matrix} \right.\$$

For $a \leq c < d \leq b$,

$$P(c \leq X \leq d) = \frac{d - c}{b - a}.$$

Probability is proportional to interval length.

Mean and Variance

$$E(X) = \frac{a + b}{2},$$

$$Var(X) = \frac{(b - a)^{2}}{12}.$$

The median and all central symmetry points equal $(a + b)/2$.

The $p$th quantile is

$$x_{p} = a + p(b - a),\quad\quad 0 < p < 1.$$

The MGF, for $t \neq 0$, is

$$M_{X}(t) = \frac{e^{tb} - e^{ta}}{t(b - a)},$$

with $M_{X}(0) = 1$ by continuity.

Example 11.1

A bus arrives uniformly between 10 and 20 minutes from now. Let $X \sim U(10,20)$. The probability that the wait is between 12 and 15 minutes is

$$\frac{15 - 12}{20 - 10} = 0.3.$$

The expected wait is

$$E(X) = \frac{10 + 20}{2} = 15\text{ minutes}.$$

Linear Transformation

If $X \sim U(a,b)$ and $Y = c + dX$ with $d > 0$, then

$$Y \sim U(c + da,c + db).$$

If $d < 0$, the endpoints are reversed and should be reordered.

Example 11.2: Generating a uniform interval

If $U \sim U(0,1)$, then

$$X = a + (b - a)U$$

has distribution $U(a,b)$. This transformation is fundamental in random-number generation.

Post-Test

1. State the PDF and CDF of $U(a,b)$.
2. If $X \sim U(2,8)$, find $P(3 < X < 5)$.
3. For Question 2, find the mean and variance.
4. Find the 90th percentile of $U(10,30)$.
5. If $U \sim U(0,1)$, state the distribution of $Y = 5 + 4U$.

Selected answers: 2. $1/3$; 3. $5$ and $3$; 4. $28$; 5. $U(5,9)$.

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