Explanations

An explanation on how the parameters change the distribution of numbers generated

In a normal distribution with the given parameters \(\sigma\), \(\mu^2\)

\[ X \sim N(\mu, \sigma^2) \]

The probability of each value is given by:

\[ P(X = x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}} \]

The code picks random values of \(0 < x < 1\), and as the equation above weighs the numbers towards the mean μ and the spread around the mean is changed by the variance \(\mu^2\). This is similar to all of the other types of distributions but with different equations to calculate the \(P(X=x)\) and therefore different steepness of curves and how affected they are by their parameters

This is easily visualized by plotting a PDF of the chosen distribution with the library, however to speed up the plotting the graph is made with systematic choices of x rather than random ones as otherwise it was too slow for large n values

A brief overview of quantiles

A quantile is a value at which, for any of the distributions X

\[P(X ≤ x) = p\ \text{for}\ 0 < p < 1 \]

This can be visualized as the area under the PDF curve, it is useful to think of it that was as when you generate a sample of random numbers with our library, you can say for sure what is the probability that we have so many values less than or equal to a certain value, which can be very useful to know. This is also shown in the 66, 80, 93, 95 rule