A uniform distribution U(a, b), also called a rectangular distribution, is defined by two parameters:
- minimum, a.
- maximum, b.
In between the minimum and maximum, the distribution has the same function values (i.e., y-values) for each input of x.
Continuous vs. discrete uniform distribution
The continuous distribution has an infinite number of values between the minimum a and maximum b. For example, you could have data points of 2.0, 2.001, 2.00001, ….
The above graph is a rectangle, so we can use the simple formula l x w to find the area under the “curve” of a continuous uniform distribution, the area is:
A = l x h = 3 * 0.333… = 1.
While there are an infinite number of continuous uniform distributions, the most commonly used is where a = 0 and b = 1 .
Another way to look at this result mathematically: the length of the rectangle is (b – a), while the height of the rectangle is 1/(b – a). Multiplying these together will always equal 1.
The continuous uniform distribution is always shaped like a rectangle – like the example above. The discrete uniform distribution is also rectangular shaped, but instead of a continuous line, a series of dots represents a known, finite number of outcomes.
As an example, one roll of a die roll has six possible outcomes: 1,2,3,4,5, or 6. There is an equal probability for each number (1/6). Discrete uniform distributions have little real-life utility and so aren’t as popular.
Continuous uniform distribution properties
Some general properties:
- Random variables can take on any value between a and b.
- Any equal size interval X is equally likely. For example, the interval [1, 3] is as equally likely as the interval [2, 4].
The general formula for the probability density function (pdf) for the continuous uniform distribution is:
Where a and b are two constants.
The uniform cumulative distribution function (CDF) is
The CDF adds up all of the probabilities from left (minimum) to right (maximum) and plots a linear graph:
Other properties include:
- Mean = ½(a + b)
- Median = ½(a + b)
- Mode = any value in (a, b)
- Skewness = 0
- Kurtosis = -6/5
Expected Value & Variance
The expected value (or mean) of a uniform random variable X is:
E(X) = (1/2) (a + b)
E(X) = (b + a) / 2.
For example, with a = 2 and b = 4, the expected value is E(X) = (4 + 2) / 2 = 3.
The variance of a uniform random variable is:
Var(x) = (1/12)(b – a)2
For example, with a = 2 and b = 4, the variance is Var(x) = (1/12)(4 – 2)2 = 1/3.
Joint uniform distribution
A joint uniform distribution can be defined relatively simply as 
For example, if you need to find the probability of a subset B of a uniform region, set up a double integral. As the density is constant, it can be pulled out, which means that the probability of a region is the ratio of its area to the sample space area:
 Penn State Eberly College of Science. Stat 414 Introduction to probability theory. 14.6 Uniform Distributions. Retrieved 4/24/2023 from: https://online.stat.psu.edu/stat414/lesson/14/14.6
 Schrader, R. Uniform distribution. Retrieved April 24, 2023 from: https://www.math.tamu.edu/~todd.schrader/419_lectures_20a/419_Uniform.pdf