Uniform Distribution

Continuous random variables are a fundamental concept in probability. Among them, the Uniform distribution stands out as the simplest. This distribution describes a situation where all possible outcomes within a given range are equally likely. For example, when generating a random number between 0 and 1, any value in that interval has the same chance of occurring.

Continuous Uniform Distribution

The continuous Uniform distribution is defined over a specific interval $[a, b]$, where $a$ is the minimum possible value and $b$ is the maximum possible value. A random variable $X$ following this distribution is denoted as $X \sim U(a, b)$.

Probability Density Function (PDF)

Unlike discrete distributions which use a Probability Mass Function (PMF), continuous distributions use a Probability Density Function (PDF), denoted $f(x)$. The PDF doesn't give the probability of a specific value (which is technically zero for any continuous variable), but rather the density of probability around that value. The probability of $X$ falling within a certain range is found by calculating the area under the PDF curve over that range.

For the Uniform distribution $U(a, b)$, the PDF is constant within the interval $[a, b]$ and zero elsewhere. This constant value ensures that the total area under the curve equals 1, a requirement for any valid PDF.

The formula for the PDF is:

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

The height of the PDF is $1 / (b-a)$, which is the reciprocal of the interval's length. This constant height reflects the "uniform" nature of the distribution, where probability is spread evenly across the interval.

The Probability Density Function (PDF) for a $U(2, 8)$ distribution. The density is constant at $1/(8-2) = 1/6 \approx 0.167$ between $x=2$ and $x=8$, and zero elsewhere. The total area under the curve is $(8-2) \times (1/6) = 1$.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF), $F(x)$, gives the probability that the random variable $X$ takes on a value less than or equal to $x$, i.e., $P(X \le x)$.

For the Uniform distribution $U(a, b)$, the CDF increases linearly from 0 to 1 over the interval $[a, b]$.

The formula for the CDF is:

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

At $x=a$, the CDF is $F(a) = (a-a)/(b-a) = 0$. At $x=b$, the CDF is $F(b) = (b-a)/(b-a) = 1$. Between $a$ and $b$, the probability accumulates linearly.

The Cumulative Distribution Function (CDF) for a $U(2, 8)$ distribution. It shows the probability $P(X \le x)$ increasing linearly from 0 at $x=2$ to 1 at $x=8$.

Mean and Variance

The expected value (mean) and variance are important summary statistics for any distribution.

• Mean (Expected Value): The mean of a $U(a, b)$ distribution is the midpoint of the interval: $E[X] = \frac{a+b}{2}$ This makes intuitive sense, as the probability is distributed symmetrically around the center of the interval.

• Variance: The variance measures the spread of the distribution. For $U(a, b)$, it is: $Var(X) = \frac{(b-a)^2}{12}$ The variance depends only on the width of the interval $(b-a)$. A wider interval leads to a larger variance, indicating greater dispersion of possible values.

Properties and Applications

The Uniform distribution's primary characteristic is that it represents complete uncertainty about the value of a variable within a known range $[a, b]$, assuming all values in that range are equally plausible.

• Random Number Generation: Most programming languages provide functions to generate pseudo-random numbers following a $U(0, 1)$ distribution. These are often the foundation for generating random samples from other, more complex distributions using techniques like inverse transform sampling or rejection sampling.
• Simulations: It's useful in simulations where an input parameter is known to be within a range, but its exact value or distribution within that range is unknown or assumed to be uniform (e.g., simulating arrival times within a certain window).
• Priors in Bayesian Statistics: In Bayesian inference, if we have no prior knowledge favoring any particular value of a parameter within a range $[a, b]$, we might assign it a $U(a, b)$ prior distribution. This is sometimes called a "non-informative" prior, though its properties require careful consideration.
• Machine Learning Initialization: Sometimes, the initial weights in neural networks are sampled from a uniform distribution over a small interval around zero, like $U(-c, c)$.

It's also worth noting the existence of the Discrete Uniform Distribution, where there are a finite number $n$ of outcomes, each having the same probability $1/n$. A classic example is rolling a fair six-sided die, where each outcome $\{1, 2, 3, 4, 5, 6\}$ has a probability of $1/6$. Our focus here, however, is on the continuous version, which is more frequently encountered when modeling continuous quantities in machine learning contexts.

The continuous Uniform distribution, while simple, serves as a fundamental building block in probability theory and its applications, particularly in simulation and as a basis for understanding random processes where outcomes over a range are equally likely. In the upcoming sections, we will explore distributions like the Normal and Exponential, which model phenomena where outcomes are not equally likely.