EE 381 Assignment 03 – Binomial and Poisson Distribution solution

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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