Uniform Distribution

Continuous Uniform

Definition

$X \sim \text{Unif}[a, b]$ has constant density over an interval.

PDF

$f(x) = \begin{cases} \frac{1}{b-a} & a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$

CDF

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

Parameters

• $a$: lower bound
• $b$: upper bound ($b > a$)

Expectation and Variance

$E[X] = \frac{a + b}{2}$

The mean is the midpoint.

Proof:

$E[X] = \int_a^b x \cdot \frac{1}{b-a} \, dx = \frac{1}{b-a} \left[ \frac{x^2}{2} \right]_a^b = \frac{b^2 - a^2}{2(b-a)} = \frac{(b-a)(b+a)}{2(b-a)} = \frac{a+b}{2}$

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

Proof:

$E[X^2] = \int_a^b x^2 \cdot \frac{1}{b-a} dx = \frac{b^3 - a^3}{3(b-a)} = \frac{a^2 + ab + b^2}{3}$

$\text{Var}(X) = E[X^2] - E[X]^2 = \frac{a^2 + ab + b^2}{3} - \frac{a^2 + 2ab + b^2}{4} = \frac{b^2 - 2ab + a^2}{12} = \frac{(b-a)^2}{12}$

Visualisation

The PDF is a flat rectangle of height $\frac{1}{b-a}$ and width $b-a$. Area = 1.

The CDF is a straight line from $(a, 0)$ to $(b, 1)$.

Examples

Example 1: Random Time

A bus arrives uniformly at random between 8:00 and 8:30. $X \sim \text{Unif}[0, 30]$ (minutes past 8:00).

$E[X] = 15 \text{ minutes}$

$P(X \leq 10) = \frac{10}{30} = \frac{1}{3}$

Example 2: Random Point

Choose a point uniformly on $[0, 1]$. $X \sim \text{Unif}[0, 1]$.

$P(X \leq 0.4) = 0.4$

$P(0.2 \leq X \leq 0.7) = 0.5$

Generating Uniform Variates

Uniform random variables are the foundation of simulation. From $U \sim \text{Unif}[0, 1]$, we can generate many other distributions.

Inverse transform method: If $F$ is a CDF with inverse $F^{-1}$, then $X = F^{-1}(U)$ has CDF $F$.

Example: Generating Exponential

For exponential with rate $\lambda$:

$F(x) = 1 - e^{-\lambda x}$

$F^{-1}(u) = -\frac{1}{\lambda} \ln(1 - u)$

So $X = -\frac{1}{\lambda} \ln(1 - U)$ or equivalently $X = -\frac{1}{\lambda} \ln U$ (since $U$ and $1-U$ have the same distribution) generates an exponential.

Properties

Probability of Interval

For $X \sim \text{Unif}[a, b]$:

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

Probability is proportional to interval length.

Memoryless?

No. For uniform, knowing $X > c$ tells us $X > c$ is impossible if $c < a$, and restricts $X$ to $[c, b]$ if $a \leq c < b$.

Linear Transformation

If $X \sim \text{Unif}[a, b]$ and $Y = cX + d$ ($c > 0$), then $Y \sim \text{Unif}[ca + d, cb + d]$.

Summary Table

Property	Value
PDF	$f(x) = \frac{1}{b-a}$ on $[a,b]$
CDF	$F(x) = \frac{x-a}{b-a}$ on $[a,b]$
Mean	$\frac{a+b}{2}$
Variance	$\frac{(b-a)^2}{12}$
SD	$\frac{b-a}{\sqrt{12}}$