8. The party starts

• Jack is organising a party at 10 pm, inviting 10 friends. He has invented a special game that involves at least 8 people (besides him). Everyone is excited, so as soon as the 8th friend arrives, the game starts. Let's say that your friends arrive independently from each other between 10 and 11 o'clock in a uniform distribution (including the fact that no one arrives before 10:00 and everyone arrives before 11). Denote \(X\) the fraction of the time interval between 10 and 11 at which the game starts (ie. \(X\) is a random variable with value between 0 and 1, so \(X=0.75\) means, the party starts at 10:45).
  1. Write a function simulating \(X\)!
  2. Consider the following two density functions: \begin{align*} f_a (x) = \frac{10!}{7! \cdot 2!}x^7 (1-x)^2, \quad 0\le x \le1,\\ f_b (x) = \frac{10!}{6! \cdot 3!}x^6 (1-x)^3, \quad 0\le x \le1. \end{align*} Simulate 3000 times the value of \(X\), draw the histogram with 50 columns, draw the two density functions, and then decide which of them may be a density function of \(X\).
  3. Using the previously generated 3000 data, substitute each simulated value of \(X\) into both density functions and multiply the substituted value by a uniformly distributed random number from \([0,1]\) (independently generated from each other). Finally, these values as the function of the simulated values are drawn in a figure. For the two density functions we get two diagrams in this figure, they look like "cloud of points." Decide which density function may belongs to \(X\) based on these two diagrams. Give a short explanation of your answer in a comment of the code.
• The output of the job is two figures. On the first there are two graphs of functions defined on \([0,1]\) and a histogram of 50 columns in the same window. The second figure consists of two subfigures, the top shows a cloud of points covering the area below the first density function, the lower shows a cloud of points covering the area below the second density function. Do not forget to add your explanations in a comment.
• A little help:
  1. You may use the next code for drawing graphs (other good solutions are possible!):

     import numpy as np
     import matplotlib.pyplot as plt
     
     # t will be a list with elements 0, .02, .04,..., 1 
     t = np.linspace(0, 1, 51)
     plt.plot(t, dfunc(t), color = "red")
     

     Here the first argument of the function plot is the list of the values of the domain, and the second argument contains the values of the function at these points, where dfunc have to be defined before.
  2. Drawing a histogram means drawing columns over intervals the height of which are proportional to the number of simulated values in the interval. There is a simple Python function to draw this histogram, where experiments is the list of the values of \(X\), bins is a list of values in the domain, density means the total area of the columns:

     plt.hist(experiments, bins=t, density=1)
     

  3. To draw points, you need to specify the style as the third argument of the function plt.plot, in this task it can be "g.", which means that the specified points will not be joined together, and the color of the points will be green (g = green).
• The aim of this exercise is to demonstrate the density function and simulate a given distribution (it is in this task the beta distribution, but it is not the part of the exercise to know or to understand this distribution!). Programming objectives are elementary programming and graphic skills.