1. Experimental Bernoulli Trials Consider the following experiment: You have three identical multi-sided unfair dice. The probability vector for the dice has been provided to you. One roll is Considered \"success \" if you get: \"one" for the first die; "two\" for the second die, "three\" form third die. You roll the three dice n:1000 times, and the number of suc :esses in [1 rolls, will be your random variable \"X\". This is considered one experiment. The goal is to create the PMF plot of "X\". - In order to generate the PMF plot repeat the experiment N =10,000 times, and record the values of "X\" each time, i.e. the number of "successes'T in 11 rolls. i Create the experimental Probability Muss Function plot, using the histogram of \"X\" as you did in previous projects. 0 Include the PMF plot in your report, in addition to all other requirements. See Figure 1 for an example of a properly labeled PMF plot. 2. Calculations using the Binomial Distribution In this problem you will use the theoretical formula for the Binomial distribution to calculate the probability p of success in a single roll of the three dice. 0 Use the Binomial formula to generate the Probability Mass Function plot of the random variable X: {number of successes in 11 Bernoulli trials }. 0 Compare the plot you obtain using the Binomial formula, to the plot you obtained from the experiments in Problem 1. 0 \"Include the PMF plot in your report, in addition to all other re .1uirements. The graph should be plotted in the same scale as the graph in Problem 1 so that they can be compared. The title should reflect the calculations for problem 2: Bernoulli Trials: PMF - Binomial .iorrnula \" 3. Approximation of Binomial by Poisson Distribution Consider the case when the probability p of success in a Bernoulli trial is small and the number of trials .11 is large (in practice this means that n 2 50 and up at 5 ). In that case you can use the Poisso11_d_i_st_1_'_i_b_ution formula to approximate the probability of success in ntrials, as an alternativeto the Binomial formula. The parameter 21 that is needed for the Poisson distribution is obtained from the equation xl : up 0 Use the parameter xiand the Poisson distribution formula to create a plot of the probability distribution function approximating the probability distribution of the random variable X = {number of successes in n Bernoulli trials }. 0 Compare the plot you obtained from the Poisson formula to the plot you obtained from the experiments in Problem 1. 0 Include the PMF plot in your report, in addition to all other requirements. The graph should be plotted in the same scale as the graph in Problem 1 so that they can be compared. The title should reect the calculations for problem 3: \"Bernoulli Trials: PMF - Poisson Approximation\
