9. Convolution of dice with discrete U-shape distributions

• A biased dice gives the number \(1\), \(2\), \(3\), \(4\), \(5\), \(6\) appropriately with probability \(0.30\), \(0.10\), \(0.05\), \(0.05\), \(0.15\), \(0.35\). Throw some dice and calculate the sum of the numbers on them. Write a program which calculates (not simulates) the probabilities of these sums of one, two, three, four, ten, and twenty dice. After this, draw these six distribution functions. In the figure for the sum of the twenty dice, draw the density function of the normal distribution well approximating this sum with a different color. Put these 6 figures in one window organizing them into two lines and three columns.
• Help: The distribution of the convolution of \(k\) dice can be determined from a set of \(6^k\) elements, but such a program runs out of time. My advice: calculate the convolution of the distribution of \(k\) dice recursively. First calculate the convolution of two dice from a diagonally summarized matrix. This distribution will have \(11\) probabilities. For 3 dice, this distribution will be convoluted with the distribution of a dice. For this you may diagonally sum the elements of a \(6\times11\) matrix. And so on.
• The file to be uploaded should be called convolution.py.
• The module numpy is useful for handling vectors and matrices.
• For grading purposes the function main() should return the object of type pyplot used for plotting. For example:

  import matplotlib.pyplot as plt
  import numpy as np
  
  ...
  
  def main():
      fig = plt.figure(figsize=(18, 9))
  
      ...
  
      return fig
  
  if __name__ == "__main__":
      main()
  

• The aim of this exercise is the calculation of convolution, and visualization of the Central limit theorem (with a distribution does not look similar to a normal distribution).