Download MATH371 – Introduction to Probability and Statistics

Take-Home Assignment 3
MATH371 – Introduction to Probability and Statistics
Fall 2005 – Dr. Lunsford
Directions: Please turn in a typed document clearly showing your answers to the
questions below (feel free to cut and paste the questions into your document). Make sure
you clearly describe your methods and results, including the pseudo code (i.e. logic) you
used to compute your simulated values of random variables. You should also place
histograms and other graphics in your document near where you describe their features.
In addition to your write-up, please turn in a disk with your Excel file (or other
appropriate file if you generated your data via some other program). You may work with
one partner on this assignment. Please turn in one write-up with both of your names and
pledges on it.
Let X equal the larger outcome when a pair of four-sided dice is rolled. The p.m.f. of X
2x 1
is f ( x) 
, x  1, 2,3, 4 . Please answer the following:
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(a) Derive, using basic principles, the p.m.f. of X .
(b) Find the mean,   E  X  , variance,  2 , and standard deviation,  , of X .
Now we are going to simulate this random experiment using two different methods:
(c) First we will simulate the fair dice. Create two columns of 100 uniform random
numbers (from zero to one). Now use those columns to simulate each die roll (i.e. you
should be able to generate two columns, one for each die, from the uniform data
columns.). Lastly, create a column for the values of the random variable X (Hint: You
will want to use Excel’s built in Max function).
 Create a histogram of your empirical p.m.f. for X (Note: Since the data is
discrete, you may want to use the qualitative histogram option in Megastat).
Compare your empirical p.m.f. to the theoretical p.m.f.
 Find the average value of your simulated values of X . This average value is an
empirical estimate of  , the theoretical mean you computed in part (b).
Compare these two values.
 Find the standard deviation of your simulated values of X . This number is an
empirical estimate of  , the theoretical standard deviation you computed in part
(b). Compare these two values.
(d) Next we will directly use the p.m.f. of X to simulate the experiment. Use the inverse
c.d.f. method discussed in class to generate 100 simulated values of X .
 Create a histogram of your empirical p.m.f. for X (Note: Since the data is
discrete, you may want to use the qualitative histogram option in Megastat).
Compare your empirical p.m.f. to the theoretical p.m.f.


Find the average value of your simulated values of X . This average value is
another empirical estimate of  , the theoretical mean you computed in part (b).
Compare these three values (your two empirical estimates and the theoretical
value).
Find the standard deviation of your simulated values of X . This number is
another empirical estimate of  , the theoretical standard deviation you computed
in part (b). Compare these three values (your two empirical estimates and the
theoretical value).
What we have done in parts (c) and (d) is to run this random experiment 100 independent
times. Thus we have taken 100 independent random samples from the distribution of X .
This is called a random sample of size 100 from the distribution of X . For each of these
random samples, we computed the average value of our simulated values of X . Each of
these averages is the sample mean of the random sample. Notice that the values of your
sample means are (most likely) not equal but should be “close” to the value of your
theoretical mean computed in part (b). The point of this is that the value of the sample
mean will vary from sample to sample (for a fixed sample size). Thus the sample mean is
also a random variable! We denote this random variable by the symbol X . The
distribution of a random variable that is obtained via a random sampling is called a
sampling distribution. In this case we have the sampling distribution of the sample mean.
In the last part of this assignment, we will get an empirical sampling distribution of X for
random samples of size 100. The theoretical sampling distribution of X is the
distribution of the sample means from all possible random samples of size n (in our case
n  100 ) from the distribution of X .
(e) Before you work the remaining part of this assignment, make a guess about the
answers to the following questions:
 How will the mean of the sample means, i.e.  X  E  X  compare to the


theoretical mean of X , i.e. E[ X ] , computed in part (b) above?
How will the standard deviation of the sample means, i.e.  X , compare to the
theoretical standard deviation of X , i.e.  , computed in part (b) above?
How will the shape of the distribution of the sample means compare to the shape
of the p.m.f. of X ?
(f) Now use the method of part (d) to create 110 random samples each of size 100 from
the distribution of X (Hint: Create your 110 columns of uniform random variables first
and then use those to create your 110 columns of simulated values of X ). For each of
those 110 random samples, compute the sample mean (use the Average function in
Excel). Thus you should have a row of 110 sample means below your row of simulated
values of X .
 Make a histogram of your simulated sample means (you may have to transpose
this row into a column first via the “paste special” option). This histogram shows
an empirical distribution for X . Compare this histogram to the p.m.f. of X . Are
you surprised?

Find the mean and standard deviation of your sample means. These are empirical
estimates of  X  E  X  and  X . Compare these values to the theoretical mean
and standard deviation of X you computed in part (b). What do you find
particularly interesting?
The purpose of parts (e) and (f) of this assignment is to introduce you to the powerful
concept of “averaging reduces variation.” The sampling distribution of X describes how
X varies about its mean and thus gives us an indication of how much error we will have
when we use a value of X obtained via a random sample of size n from a distribution
with mean  and standard deviation  in order to estimate  . We will be examining
these ideas in detail when we cover sampling distributions and the Central Limit
Theorem!