## Description

0. Introduction and Background Material

0.1. Random experiments that can be described by well-known probability

distributions

In this project you will simulate the rolling of three dice n times. Your random

variable “X” is the number of “three sixes” in n rolls. This is considered one

experiment. You will repeat the experiment N times and you will create the

probability distribution of the variable “X”.

As an alternative method to the simulation experiments, you will use the formula

for the Binomial distribution to calculate the probability distribution for the

random variable “X”. This method involves only calculations using the binomial

formula, and does not involve simulations.

Similarly, another alternative to the simulation experiments, is to use the

formula for the Poisson distribution, which can approximate the Binomial under

certain conditions.

0.2. Binomial distribution

Consider the following experiment: You toss a coin, with probability of success p and

probability of failure q p = −1 . This toss is called a Bernoulli trial. You repeat

tossing the coin n times, i.e. you have n Bernoulli trials. These n Bernoulli trials

are independent, since the outcome of each trial does not depend on the others. The

question is: what is the probability of getting exactly x successes in n independent

Bernoulli trials?

The answer can be calculated from the Binomial distribution: consider the random

variable X = {number of successes in n Bernoulli trials}. Then:

EE 381 Project : Binomial & Poisson Distributions Dr. Chassiakos – Page 2

( ) x nx n

pX x pq

x

− = =

The probability distribution of X in called the Binomial distribution.

0.3. Poisson distribution

Consider the following experiment: You observe the occurrence of a particular event

during a time interval that has duration one unit of time. You count how many

times the event has occurred during this interval. The occurrences are independent

of each other, and the event occurs at an average rate of λ times per unit of time.

The question is: what is the probability of getting exactly x occurrences during the

observation interval (which has duration of one time unit) ?

The answer can be calculated from the Poisson distribution: consider the random

variable X = {number of occurrences during a unit time interval}. Then:

( ) !

x

e pX x

x

λ λ −

= =

EE 381 Project : Binomial & Poisson Distributions Dr. Chassiakos – Page 3

1. Experimental Bernoulli Trials

Consider the following experiment:

You roll three fair dice n=1000 times. This is considered one experiment, or one

Bernoulli Trial. If you get “three sixes” in a roll, it is considered “success”. The

number of successes in n rolls, will be your random variable “X”. The goal is to create

the probability mass function plot of “X”.

• In order to generate the histogram repeat the experiment N=10,000 times, and

record the values of “X” each time, i.e. the number of “successes” in n rolls.

• Create the experimental Probability Mass Function plot, using the histogram

of “X” as you did in previous projects.

• SUBMIT the PMF plot and your code in a Word file. Use 16 bins to plot the

results. All plots should be properly labeled. See Figure 1 for an example. Note:

Do not replicate the “scroll” in Figure 1. The scroll is used in the figure in order

to hide the graph data.

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. Success is

defined as the number of “three sixes” in n = 1000 trials.

• Use the Binomial formula to generate the Probability Mass Function plot of

the random variable X = {number of successes in n Bernoulli trials}.

• Compare the plot you obtain using the Binomial formula, to the plot you

obtained from the experiments in Problem 1.

• SUBMIT the PMF plot and your code in a Word file. The graph should be plotted

in the same scale as the graph in Problem 1 so that they can be compared.

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 n is large (in practice this means that n ≥ 50 and np ≤ 5). In

that case you can use the Poisson distribution formula to approximate the

probability of success in n trials, as an alternative to the Binomial formula. The

parameter λ that is needed for the Poisson distribution is obtained from the

equation λ = n p

• Use the parameter λ and 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}.

• Compare the plot you obtained from the Poisson formula to the plot you

obtained from the experiments in Problem 1.

• SUBMIT the PMF plot and your code in a Word file. The graph should be plotted

in the same scale as the graph in Problem 1, so that they can be compared.

EE 381 Project : Binomial & Poisson Distributions Dr. Chassiakos – Page 4

Experimental results

0 2 4 6 8 10 12 14 16

Number of successes in n=1000 trials

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Probability of success

Figure 1. Example of an appropriately labeled histogram.

Note:

DO NOT REPLICATE THIS SCROLL