In this exercise you have learnt how we use the central limit theorem and the law of large numbers to estimate probability mass functions and probability density functions from identically distributed and indepedent random variables. To consolidate what you have learnt write a python notebook that repeats the last exercise above - the estimation of the probability density function for the normal distribution. Do not update the graph after each new point is generated though as it will make your code very slow. Instead only plot the histogram once you have generated all the random variables. Try to work out how to calculate the cumulative probability distribution function from this probability mass function (hint: you will need to integrate the probability density function you estimated numerically). Try to do this exercise with different numbers of subranges. In your notes discuss using suitable diagrams how the shape of the cumulative probability distribution function you obtain through this procedure differs from the shape of the true cumulative probability distribution function.

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