Problem
The central limit theorem states that no matter the underlying distribution as long as you have a relatively uniform mean and variance the sum of the distributions will become normal. In our experiment we will develop our own function that creates a uniform distribution with a number of samples N composed of a number of random variables n. By observing the actual histogram of our function output we explore how as we increase the number of random variables our PDF becomes more normal. We plot the actual mean, variance, and MSE between the actual pdf and a normal fitted distribution to observe the underlying characteristics of our output.
Activity four of the assignment tasks us with creating a new method for creating a normal distribution of random variables using the Box-Muller technique. To compare the two functions we develop we time and compare the speed and accuracy of our two approaches for creating normal distributions.
Approach and Results
To begin we use our uniform generating function to create a single uniform distribution. The resulting plot is obviously uniform and does not fit a normal distribution too well. A single uniform distribution had the highest mean squared error out of all the plots (we will look at the MSE later in the assignment). Next we increase the number of random variables (n) to ten and observe whether we see whether we can see application of the central limit theorem.
At ten random variables we see a pretty normal distribution from the sum of the samples returned using our uniform generating function. We extrapolated from our MSE plot (Figure 1) a MSE of 0.01. Also note how the possible range of data has increased from [-1,1] to [-10, 10]. Next we will increase the number of random variables to one hundred and observe the actual vs normal fitted PDF.