S4 Approximation by Discrete Uniforms

Intuitively, the Uniform random variable U[a,b] is equally likely to take all values within the interval [a,b]. To be more precise, the probability of taking a value within any subset S of the interval [a,b] is proportional to the size of the set S within [a,b]. Suppose a=1 and b=3. Then the set, [1,2] is half of the full set [1,3] and random variable U[1,3] would take values in this set with probability 50%. Every continuous random variable can be well approximated by a discrete one. For example, suppose random variable V takes values 1, 1.5, 2, 2.5, 3 each with value 20%. Since these numbers are evenly spaced and equi-probable, the variable V is a crude approximation of the uniform random variable U. To get better approximations, we need to refine the grid of values. For example, if we consider 1.0, 1.1, 1.2, …, 2.9, 3.0 as 21 equi-probable outcomes of a random variable W, then W is a one-decimal place accurate approximation to U. We can continue this process by consider 1.00, 1.01, 1.02, …. ,2.98, 2.99, 3.00 as 201 point approximate to U. This will provide us with a two decimal place approximation to the continuous random variable.

The graph below shows a FIVE point discrete approximation to the Uniform distribution between -1 and +1. Consider a discrete random variable D with takes values -0.8, -0.4, 0, 0.4, 0.8 with equal probability 1/5 each. These five points are evenly spaced between -1 and +1. We plot the CDF of the five point discrete distribution together with the Uniform CDF, and note that the two are close to each other. The maximum difference between the two graphs is 0.1 or 10%. The maximum difference between the CDF is one measure of how much random variables differ from each other.

We can do much better with a 10 point approximation, which will be accurate to about 1 decimal place. This involve taking ten points -0,9 -0.7 -0.5 -0.3 -0.1 +0.1 0.3 0.5 0.7 0.9 and constructing a discrete random variable which gives equal probability 1/10 to each one of the them. Below we plot the CDF of this approximation together with the CDF of the uniform to show how close they are. This time the maximum discrepancy between the two CDF is 0.05.