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FACULTY OF SCIENCE
DEPARTMENT OF STATISTICS
STATISTICS 1A
STA01A1
DISTRIBUTION THEORY
LEARNING GUIDE
i
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Welcome
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Welcome to Statistics 1A!
Purpose of the Module
To provide the student with a perspective of the basics of probability theory and to illustrate
its application to the solution of practical problems. The student will also be given a
fundamental perspective of a variety of discrete probability distributions and will be able to
apply them to solve problems in various fields of application.
Learning Outcomes of the Module
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➢
Define and apply the basic concepts of probability theory
Define and apply random variables, their joint distributions and probability densities
Module Assessment Criteria
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Write down a discrete sample space corresponding to the description of a random
experiment
Calculate probabilities of events by enumeration of points in the sample space
Calculate probabilities of compound events using probabilities given for simple events
using various techniques such as partitioning of the sample space and conditional
probability
Define and apply the concept of a discrete random variable, its distribution its and
probability function
Use histograms to evaluate experimental measurements and apply the chi-square
test
Define binomial and Poisson random variables, state their probability functions,
evaluate associated probabilities and determine the mean and variance of these
random variables
Explain the concepts of independence, jointly distributed discrete random variables
and establish the mean and variance of linear combinations of independent discrete
random variables
Course Map
Learning Unit 1
Overview and Descriptive Statistics
Learning Unit 2
Probability
STATISTICS 1A
Learning Unit 3
Discrete Random Variables and
Probability Distributions
Learning Unit 4
Continuous Random Variables and
Probability Distributions
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Quick Help
Where to start:
Check your learning package to ascertain that you have the following:
•
•
The learning guide for Statistics 1A STA01A1
A general information sheet
For assistance logging onto the student portal, go to http://student.uj.ac.za or ask the
assistants in the computer laboratory to help you.
•
From the student portal, go to uLink to explore the different tools in your web
learning environment for important information:
➢
General Information
➢
Course Material
➢
Announcements
➢
Assessment times and venues
➢
Assessments
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Table of Contents
Quick Help ................................................................................................................ iv
Section A: Administrative details ............................ Error! Bookmark not defined.
Section B: Support Services and Resources ........................................................ 5
Section C: Facilitation of learning .......................................................................... 7
Learning unit 1: Overview and Descriptive Statistics ........................................... 7
Learning unit 2: Probability .................................................................................. 9
Learning unit 3: Discrete Random Variables and Probability Distributions ........ 10
Learning unit 4: Continuous Random Variables and Probability Distributions ... 11
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Section A:
Administrative
details
LECTURER:
Dr J Van Appel
Email:
jvanappel@uj.ac.za
ASSESSMENTS:
The six assessments mentioned in the table below will make up the semester mark
(SM) for Statistics 1A as follows:
The best two of the three marks for Assignments 1, 2 and 3:
2 × 16% = 32%
The best two of the three marks for Summative Assessments 1, 2 and 3:
2 × 34% = 68%
100%
Assessments
Date
Assignment 1
4 March 2024
Assignment 2
15 April 2024
Assignment 3
13 May 2024
Summative Assessment 1
11 March 2024 (T)
Weighting
16%
14 March 2024 (P)
22 April 2024 (T)
Summative Assessment 2
25 April 2024 (P)
20 May 2024 (T)
Summative Assessment 3
23 May 2024 (P)
2
34%
For example, if you miss Assignment 1, you can still do Assignments 2 and 3, and
those 2 marks will then each contribute 16% towards your semester mark. If you
submit all three assignments, the best two marks will be used as your 32%
contribution to the semester mark. If you fail to submit any assignments, you will
forfeit 32% of your semester mark. The same applies to Summative Assessments 1,
2 & 3.
You will have three weeks to complete each of the three assignments. The three
summative assessments will be timed, and you will have up to 1.5 hours to complete
the theory paper and up to 1 hour to complete the practical on the dates as given in
the table above. Each assessment will cover both theoretical and practical
applications.
The final summative assessment (“exam”) for STA01A1 will be written on 18 June
2024. You must obtain a semester mark (SM) of at least 40% to qualify for this final
assessment. The final mark (FM) for STA01A1 will be calculated as follows:
50% × 𝑆𝑀 + 50% × 𝐸𝑀 = 𝐹𝑀
where EM = mark for the final summative assessment on 18 June 2024 (“Exam Mark”).
You need a subminimum EM of 40% and an FM of at least 50% to pass Statistics
1A.
The three assignments will be online. All the summative assessments will be
written ON CAMPUS. Please note these dates and make arrangements to attend the
on-campus assessments.
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Section B:
Support Services
and Resources
We have made the following resources available to ensure you have all the support
you need to complete this course successfully. You will do well if you follow our
suggestions for studying this course and use all these resources to their full potential.
The resources are:
1.
Learning Guide
The purpose of the learning guide is to support your learning process. It serves
as a map of material available in the different resources.
FACULTY OF SCIENCE
DEPARTMENT OF STATISTICS
LEARNING GUIDE
STATISTICS 1A
STA1A10
DISTRIBUTION THEORY
2.
Prescribed textbook
The prescribed textbook, available on miEbooks
, is:
Title:
Probability and Statistics for Engineering and the Sciences
Devore, Jay L.
ISBN:
9781305465329
It will be used for the first and second semesters and as a handy reference for
the second year.
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3.
Web learning environment: uLink
You must access uLink to retrieve all the online resources and assessments.
Please refer to the logon procedure on the uLink brochure.
4.
MERLOT
MERLOT (Multimedia Educational Resource for Learning and Online Teaching)
is a curated collection of free and open online teaching, learning, and faculty
development services contributed to and used by an international education
community.
Feel free to use this resource any time you need extra help on any specific topic
in Statistics.
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Section B:
Facilitation
of learning
Learning unit 1: Overview and Descriptive Statistics
We begin our study of statistics by noting that the word has two primary meanings. In one
sense, the word refers to specific numbers, such as “fifty-six per cent of students either
strongly or moderately agreed with the statement ‘I am afraid of Statistics’.” A second
meaning refers to statistics as a method of analysis. It is this second meaning that will
concern us in this course.
We will look at numerical quantities that describe the main characteristics of a distribution.
Graphical representations of distributions help gain a bird’s eye view of what the data are
telling us. However, a few complex numbers are needed to make scientifically sound
conclusions and comparisons between distributions. Numerical quantities that succinctly
describe a distribution's characteristics are known as statistics of the data (not to be
confused with the scientific discipline known as Statistics). We are looking to find a small
number of quantities that summarize the content of the data without throwing away essential
information.
Unit learning outcomes
At the end of this unit, you should be able to:
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Differentiate between a population and a sample
Identify a random variable
Distinguish between the branches of Statistics
Differentiate between the methods of collecting data
Construct and interpret graphic displays of quantitative data
• Stem-and-Leaf displays
• Dot plots
• Histograms
❖ Describe the shape of a histogram
❖ Construct and interpret graphic displays of qualitative data
• Bar graph
• Pie chart
❖ Distinguish between measures of central location, measures of non-central location,
measures of dispersion, and measures of skewness
❖ Compute and interpret the mean, median and mode, with particular emphasis on when
each is best used
❖ Compute and interpret percentiles and quartiles
❖ Calculate and interpret a 100α% trimmed mean
❖ Compute and compare the characteristics of each measure of central tendency
❖ Compute the range, inter-quartile range, standard deviation and variance
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❖ Calculate and interpret a 5-number summary for s data set
❖ Draw a box plot
❖ Interpret comparative boxplots
Unit resources
Textbook:
➢ Chapter 1
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Learning unit 2: Probability
How do you deal with uncertainty? By understanding how it works. Probability starts by
encouraging you to clarify your thinking to separate the genuinely uncertain things from the
hard facts. What exactly is the uncertain situation you are interested in? By what exact
procedure is it determined? How likely are the various possibilities? Often, there is some
event that either will or will not happen. Will you get the contract? Will the customer send in
the order form? Will they fix the machinery on time?
Unit learning outcomes
At the end of this unit, you should be able to:
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Understand and define an experiment, a sample space and an event
Understand classical probability and the assumptions underlying it
List the three axioms of probability
Interpret probability in terms of the relative frequency
Proof properties of probabilities
Calculate probabilities using counting techniques when the outcomes of an
experiment are equally likely
• Product rule
• Tree diagrams
• Permutations
• Combinations
Define and calculate conditional probabilities
Derive the multiplication rule
Use the multiplication rule
Formulate and prove the law of total probability
Use Bayes’ Theorem
Define two independent events
Determine P(A∩B) when A and B are independent events
Unit resources
Textbook:
➢ Chapter 2
8
Learning unit 3: Discrete Random Variables and Probability
Distributions
You can also think about random variables as where data sets come from. Random
variables generated many of the data sets you worked with in the previous units. In this
sense, the random variable represents the population (or the process of sampling from the
population), while the observed values represent sample data. The fundamentals of random
numbers are covered in this unit. The pattern of probabilities for a random variable is called
its probability distribution.
All the random variables considered in this unit have a mean and a standard deviation. In
addition, each event has a probability based on a random variable.
Unit learning outcomes
At the end of this unit, you should be able to:
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Understand the concept of a probability distribution
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Define a
• Random variable
• Bernoulli random variable
• Discrete random variable
• Continuous random variable
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Use a probability distribution in finding probabilities associated with a given value
of the random variable
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Define the expected value of a discrete random variable
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Proof the properties of expected values
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Formulate, apply and interpret Chebyshev
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Calculate moments
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Calculate a measure of skewness
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Define a moment-generating function (mgf)
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Obtain moments by using the mgf
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Understand the nature and the applications of the Binomial distribution
• Derive mgf
• Derive expected value and variance using mgf
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Understand the nature and the applications of the Hypergeometric distribution
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Understand the nature and the applications of the Negative binomial distribution
• Recognize mgf
• Derive expected value and variance using mgf
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Understand the nature and the applications of the Poisson distribution
• Derive mgf
• Derive expected value using mgf
Unit resources
Textbook:
➢ Chapter 3
9
Learning unit 4: Continuous Random Variables and Probability
Distributions
Unit learning outcomes
At the end of this unit, you should be able to:
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Define the probability distribution (pdf) of a continuous random variable
Name and verify the two conditions of a pdf
Define the Uniform distribution
Define the cumulative distribution function of the Uniform distribution
Define and obtain the expected value of a continuous random variable
Define the variance and standard deviation of a continuous random variable
Define the mgf of a continuous random variable
Use the mgf to calculate the mean and variance of a continuous random variable
Derive the mgf of the uniform distribution
Use the mgf of the uniform distribution to obtain the mean and variance
Define the Normal probability distribution
Define the cumulative distribution function of the standard normal distribution
Use the Normal tables to calculate probabilities
Formulate the Empirical rule
Derive the mgf of the normal distribution
Use the mgf of the normal distribution to obtain the mean and variance
Unit resources
Textbook:
➢ Chapter 4
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