Continuous uniform / rectangular distribution

Understanding Continuous Uniform Distribution

Continuous Uniform Distribution, also known as Rectangular Distribution, represents a situation where each value within a certain range is equally likely to be observed.
Distinct to this type of distribution is how the probability density function remains constant within the specified range, a characteristic depicted by a rectangle in a graph. Hence the term “rectangular distribution”.
Key parameters of a continuous uniform distribution are the minimum (a) and maximum (b) values that define the range. Together, they specify the area of probability.

The PDF (Probability Density Function) of a continuous uniform distribution is 1/(b - a) for a ≤ x ≤ b. For values outside this range, the probability is zero.
The CDF (Cumulative Distribution Function) calculates the probability that a random variable X will take a value less than or equal to a given value x. It’s defined as (x - a)/(b - a) for a ≤ x ≤ b.
The mean (μ) of a continuous uniform distribution is given by (a + b) / 2, which is the midpoint of the distribution range.
The variance (σ^2) is given by (b - a)² / 12. It provides the measurement of the spread around the mean.

When drawing a sample from a continuous uniform distribution, every value within the defined range has an exactly equal probability of being chosen.
This uniform probability is denoted by a horizontal line in the probability density function.
This ability to produce an equal chance of selection makes this distribution useful in several practical applications, such as random number generation.

Continuous uniform distribution allows modellers to assign equal probabilities to all the points of a certain range.
In forecasting and decision-making processes, a continuous uniform distribution might be used when there is maximum uncertainty about the likelihood of an outcome.
The simplicity of continuous uniform distribution is a key advantage. However, real-world problems are often more complex and may require different types of distribution.

The main limitation of a continuous uniform distribution is that it assumes an unrealistically constant probability for all values within a given range.
It doesn’t allow for the fact that in many real-world scenarios, certain outcomes within a range may be more likely than others.
It assumes perfect information about the minimum and maximum values, which might not always be possible.