9.1 Uniform distribution

Uniform Distribution Properties

The uniform distribution is the simplest continuous probability distribution. It models situations where every outcome in a range is equally likely. Think of a random number generator that picks any real number between 0 and 10 with no preference for any value over another.

Probability Density Function and Parameters

A uniform distribution is defined by just two parameters:

• Lower bound ($a$): the smallest possible value
• Upper bound ($b$): the largest possible value
• The only constraint is $a < b$

Because every value in the interval is equally likely, the probability density function (PDF) is flat:

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

Outside that interval, $f(x) = 0$. Graphically, the PDF looks like a rectangle, which is why you'll sometimes hear this called the "rectangular distribution."

The cumulative distribution function (CDF) gives the probability that the variable is less than or equal to $x$:

$F(x) = \frac{x - a}{b - a} \quad \text{for } a \leq x \leq b$

This is a straight line that climbs from 0 at $x = a$ to 1 at $x = b$.

Statistical Measures and Characteristics

Because the distribution is perfectly symmetric, the mean and median are the same:

$\mu = \frac{a + b}{2}$

The variance measures how spread out the values are:

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

The interquartile range (IQR) is $(b - a)/2$, which is exactly half the width of the interval.

One especially common version is the standard uniform distribution, $U(0, 1)$. It's the building block for random number generation and for a technique called the probability integral transform, which converts samples from one distribution into another.

Probabilities and Quantiles for Uniform Variables

Probability Calculations

Calculating probabilities with a uniform distribution is straightforward because the PDF is constant. The probability that a uniform random variable falls within a subinterval $[c, d]$ inside $[a, b]$ is just the ratio of the subinterval's length to the total interval's length:

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

For example, if $X \sim U(2, 10)$ and you want $P(4 \leq X \leq 7)$:

$P(4 \leq X \leq 7) = \frac{7 - 4}{10 - 2} = \frac{3}{8} = 0.375$

A few things to keep in mind:

• Any subinterval outside $[a, b]$ has probability 0, since the PDF is 0 there.
• The probability of the variable equaling any single exact value is 0. This is true for all continuous distributions, not just the uniform.
• Two subintervals of equal length within $[a, b]$ always have the same probability, no matter where they sit in the range.

Quantile Determination

Quantiles answer the reverse question: given a probability $p$, what value does the variable fall below with that probability? You find them using the inverse of the CDF:

$Q(p) = a + p(b - a)$

Here are the key quantiles for any $U(a, b)$:

Applications of the Uniform Distribution

Modeling Equal Likelihood Scenarios

The uniform distribution fits any situation where outcomes in a range are equally likely:

• Rounding errors: When a digital scale rounds to the nearest gram, the actual rounding error is roughly uniformly distributed across the rounding interval.
• Waiting times: At an automated car wash with a fixed 5-minute cycle, your wait time if you arrive at a random moment is approximately $U(0, 5)$ minutes.
• Random selection: Picking a random real number between two bounds, such as a spinner landing anywhere on a circular dial.

Computer Science and Simulation

• Random number generation: Most pseudo-random number generators produce values from $U(0, 1)$ as their starting point. Other distributions are then built from these uniform samples.
• Cryptography: Generating encryption keys and initialization vectors requires uniformly distributed random bits so that no key is more predictable than another.
• Monte Carlo simulations: These methods rely on uniform random samples to estimate complex quantities, from option prices in finance to particle behavior in physics.

Other Fields

• Manufacturing and quality control: Measurement errors within a known tolerance range are sometimes modeled as uniform.
• Insurance and actuarial science: When claim sizes are known to fall within fixed bounds but no value is more likely, a uniform model can serve as a starting assumption.
• Game theory: Mixed strategy equilibria often involve players randomizing uniformly over a set of actions.