Step 1 — Introduction to Uniform Distribution
The concept of a uniform distribution is tailored for continuous valued data. It revolves around a singular value that is denoted as 1/(b−a) and it holds significance within a specific range [a, b]. If we go, beyond this range all values are set to zero. This distribution can be likened to a categorical variable with two possible categories either 0 or the given value. Interestingly this categorical variable can assume multiple values within the continuous range defined by numbers a and b.
Mathematical representation:
Step 2 — Normalizing Frequency with Density
By utilizing the density option we can normalize the occurrence frequency of each outcome. Instead of focusing solely on frequency, this option provides us with the relative frequency of occurrences.
Step 3 — Controlling Data Groups
The bins option plays a crucial role in dictating the number of data groups in the distribution. This parameter let us control the granularity of data representation.
Step 4 — Initial Distribution Visualization
The first output image generated from the data showcases an initial approximation of the ideal uniform distribution. Due to a limited sample size of the random variable we take only 100 samples, the approximation may not be perfect. As a result, some bins or groups exhibit a relative frequency exceeding 1 while others display a relative frequency below 1.
Now we take 1000 samples and our density is same in this case that is 1 and we cannot modify any parameter, let’s see what’s the result in output.
Step 6 — Enhanced Approximation with Increased Sample Size
One noteworthy observation is that the accuracy of the approximation to the ideal uniform distribution improves as the sample size of the random variable increases. In the final output image which employs a substantial sample size of 100,000 random variable samples, the representation is notably closer to the ideal uniform distribution.
Conclusion
In essence, the uniform distribution offers a unique perspective on continuous valued data, characterized by a distinct value within a predefined range. Through the density and bins options, we gain control over frequency normalization and data grouping respectively. The distribution’s behavior and accuracy are vividly illustrated in the visual outputs, underscoring the importance of sample size in achieving a faithful representation of the ideal uniform distribution.
