Introduction
The aim of this collection is to provide lecturers and students with a battery of problems
that can be used to support basic and intermediate courses in probability and statistics.
Two types of problems are included:
1. Problems where simulation in Excel is used either to clarify probabilistic/statistical
concepts or to obtain approximate solutions to problems in probability and
statistics.
2. Problems that use real or simulated data to illustrate the use of statistical methods.
In the majority of the problems simulation is used to provide insights into concepts and/or
to compute approximate solutions to probabilistic and statistical problems. I have opted for
the use of Excel as simulation tool. This has certain drawbacks in terms of capabilities
compared to more advanced software. On the other hand it is a major tool used in many
introductory and intermediate courses in Probability and Statistics and is also readily
available to virtually every student.
In my many years of teaching such courses I have experienced the advantages of using
simulation to help beginning students grasp the concepts of Probability and Statistics.
Indeed even basic concepts or problems in this field are often a big hurdle for them to take.
Tossing two fair coins and asking for the probability that the two coins show a different
outcome will trigger quite a few wrong answers from beginning students. A simple
simulation of this experiment will convince them that their answer is either right or wrong.
Although a simulated approach to a problem is often useful this is not to say that an
analytical solution based on a correct model is redundant, quite on the contrary. In fact, in
many of the problems an analytic solution is asked for so that the approximate solution
provided by the simulation can be compared to it. This dual approach to problem solving is
often very insightful. Personally I must admit that I have come across several problems in
probability where I was only 95% convinced, or even less, that my analysis and answer was
correct. Although simulation could not always be of help, in several cases it was able to
convince me that my answer was right (or wrong). This observation certainly holds for a
(beginning) student, be it at a more elementary level.
Being familiar with simulation has a number of advantages for the (beginning) student of
Probability and Statistics which can be summarized as follows:
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creates probabilistic intuition but at the same time shows that our intuition is
sometimes severely fooled;
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helps students to overcome their fears and prejudices of the field;
gives good insight into basic concepts such as randomness, the notion of probability,
expected value, correlation, etc;
strengthens the belief that an analytical solution is correct;
provides the basis for more complicated models needed to solve real world
problems. These problems are often so messy that they are beyond the scope of an
analytical approach.
For the second type of problems Excel is used on simulated or real data sets to illustrate
statistical methods. Twelve data sets are included most of them rather small in size but
three sets have 2000 or more observations.
Occasionally the Solver of Excel is used to solve problems of either type, e.g. when solving
maximum likelihood type problems. It should be pointed out that the Solver does not work
(well) when combining with a data table (often used for simulation) in the same workbook.
This combination should be avoided.
The level of difficulty of the problems in this collection is variable. On the one hand I have
included elementary problems such as using simulation to illustrate the meaning of the
concept of probability, random variable, probability mass function and density function,
expected value, correlation, confidence interval, statistical test, etc. On the other hand the
collection also contains more advanced problems where basic real analysis and matrix
algebra are needed.
Most of the problems have been used at one time or another in the many basic courses I
have taught for two types of students. The more basic problems were used for or assigned
to second year undergraduate business students in business administration who took a first
course in probability/statistics. This group of students had a basic but not very formal
course in analysis and matrix algebra in their first year. Students in the second group
followed our so called commercial engineering program, a program which contains more
quantitative and engineering components besides the economic/business component. I
used the more advanced problems for these students. Although the problems were used in
business programs most are quite general and as such can be used for students of different
disciplines.
Every problem in the collection consists of two parts. Part 1 describes a problem, asks for
an analytical solution (if applicable or possible), then shows how to simulate the problem or
analyze the data. Detailed Excel sheets with solution are provided for this part of the
problem. I should point out that in most cases a problem can be simulated in several
different ways. Many times I have opted for the more instructive approach which is not
always the most summary approach. Part 2 of a problem is meant as an assignment for the
student. For most problems this part is along the same lines as part 1 but no answers are
provided.
The version of Excel used is version Excel 2010 (no Visual Basic is used). Changes needed
for Excel 2007 or Excel 2013 users are minimal.
Some abbreviations are used in the text:
pmf: probability mass function p(.) (of a discrete random variable)
pdf: probability density function f(.) (of a continuous random variable)
pd: (cumulative) distribution function F(.)
Whenever the term ‘random number’ is used it always means a random number between 0
and 1 as generated by the function RAND() in Excel.
The collection contains:
1. Text: the text of all the problems: 16 chapters for a total of 303 problems (Word file);
2. Excel Solutions: Excel workbooks with the solutions of all problems (except the
solutions for the assignments for students);
3. Analytics: the solutions of the analytical part of the problems in Chapters 1 to 6 and
chapter 10 (Word files);
4. Data: the 12 data sets (Excel files) and a description of the data sets (Word file).