ENGI 4421
Discrete Probability Distributions
Discrete Probability Distributions
Page 9-01
[Navidi sections 4.1-4.4; Devore sections 3.4-3.6]
Chapter 5 introduced the concept of probability mass functions for discrete random quantities.
The only standard d

ENGI 4421 Probability and Statistics
Faculty of Engineering and Applied Science
Problem Set 3 Solutions
Probability; Counting Techniques
In questions 1-4, evaluate manually and show your working:
1.
10
C3
10
C3 =
10 9 8 7!
3 2 1 7!
= 5 3 8
= 120
2.
10
P3

Term Test 2 Solutions 2012 06 25
ENGI 4421
1)
Page 1 of 6
A storeroom contains 25 air compressors that were all assembled in the same
production run. It is known that five of those air compressors would fail a quality
control test. From the storeroom a ra

ENGI 4421
1)
Term Test 1 Solutions 2011 05 30
Page 1 of 4
The masses (in grammes) of ten randomly selected corroded bolts in a pipeline are
listed here, in ascending order.
1.75
8.73
9.58
10.81 10.89 11.42 11.80 12.03 12.96 14.23
(a) Find the value of the

ENGI 4421
Introduction to Probability; Sets & Venn Diagrams
Page 2-01
Probability
1
u1
1
2
u2
2
1
u3
2
u4
Decision trees
Decision
Actions
Chance nodes
Consequences;
utility
States of nature
Which action will be taken is a decision completely controlled b

ENGI 4421
Propagation of Error
Propagation of Error
Page 8-01
[Navidi Chapter 3; not in Devore]
Any realistic measurement procedure contains error.
Any calculations based on that measurement will therefore also contain an error.
The errors are propagated

ENGI 4421
Discrete Random Quantities
A random quantity [r.q.] maps an outcome to a number.
Page 5-01
[Navidi Section 2.4]
[Devore Sections 3.1-3.3]
Example 5.01:
P = A student passes ENGI 4421
F = That student fails ENGI 4421
The sample space is S = cfw_

ENGI 4421
Example 3.01
Counting Techniques for Probability
Page 3-01
[Navidi Section 2.2; Devore Section 2.3]
Four cards, labelled A , B , C and D , are in an urn.
In how many ways can three cards be drawn
(a) with replacement?
(b) without replacement?
(c

ENGI 4421
Discrete Probability Distributions
Discrete Probability Distributions
Page 9-01
[Navidi sections 4.1-4.4; Devore sections 3.4-3.6]
Chapter 5 introduced the concept of probability mass functions for discrete random quantities.
The only standard d

ENGI 4421
Descriptive Statistics
A Brief Definition of Statistics:
Page 1-01
[Navidi Chapter 1 & Devore Chapter 1]
Statistics is the science of making decisions in the absence of certainty. .
Some more definitions:
A population is a set of objects.
A samp

ENGI 4421
Central Limit Theorem
Page 11-01
Central Limit Theorem [Navidi, section 4.11; Devore sections 5.3-5.4]
If Xi is not normally distributed, but E[Xi] = , V[Xi] =
(approximately 30 or more), then, to a good approximation,
2
and n is large
2
X ~ N

ENGI 4421
Continuous Probability Distributions
Example 6.01:
Page 6-01
[Navidi section 2.4; Devore sections 4.1-4.2]
Exact lifetime is a continuous random quantity, but
Measured lifetime to the nearest minute is a discrete random quantity.
In a bar chart,

ENGI 4421
Chi-Square Tests
The Chi-Square Goodness-of-Fit Test
Page 14-01
[Navidi section 6.10; Devore chapter 14]
One simple test to determine whether or not a given random sample is consistent with some
probability distribution is to compare the numbers

ENGI 4421
Hypothesis Tests
Page 13-01
Hypothesis Tests
[Navidi sections 6.1-6.8 and 6.12-6.13; Devore chapter 8 and sections 9.1-9.4]
We begin with a reminder of page 7-14:
Is a null hypothesis Ho true (our default belief), or do we have sufficient eviden

ENGI 4421
Continuous Probability Distributions
Page 10-01
Continuous Probability Distributions
[Navidi sections 4.5-4.8 and 4.10; Devore sections 4.3-4.5]
Chapter 6 introduced the concept of probability density functions for continuous random
quantities.

ENGI 4421
Confidence Intervals (Two Samples)
Page 12-01
Two Sample Confidence Interval for a Difference in Population Means
[Navidi sections 5.4-5.7; Devore chapter 9]
From the central limit theorem, we know that, for sufficiently large sample sizes from

ENGI 4421
Conditional Probability and Independence
Example 4.01
Page 4-01
[Navidi Section 2.3; Devore Sections 2.4-2.5]
Given that rolling two fair dice has produced a total of at least 10, find the probability
that exactly one die is a 6.
Let A = "total

ENGI 4421
Simple Linear Regression
Page 15-01
Simple Linear Regression
[Navidi Sections 7.2-7.4; Devore Chapter 12]
[This topic is treated somewhat differently here from the approach in the textbook.]
Sometimes an experiment is set up where the experiment

ENGI 4421
1)
Term Test 2 Solutions 2013 07 03
Page 1 of 6
The mass M of a bolt from a production line is known to be a normally distributed
random quantity with mean = 5.0 g and standard deviation = 0.3 g . The mass
of each bolt is independent of the mass

ENGI 4421
1)
Term Test 1 Solutions 2012 05 28
Page 1 of 7
The histogram displays fifty observations of the completion time t in hours for one
phase of an engineering contract.
(a) From the histogram calculate the number of observations in the class 1.4 t

ENGI 4421
Statistical Tables
Page 17-01
Appendix to the Lecture Notes
for ENGI 4421 Probability and Statistics
Some Statistical Tables
On the next five pages are the three most frequently used statistical tables:
critical values of the t distribution;
val

ENGI 4421
Suggestions for Formula Sheets
Page 16-01
Some formul for ENGI 4421 Probability and Statistics
On these pages are many more formulae than will fit comfortably on your allocation of
two sheets in the final examination. Examine these suggestions t

ENGI 4421
Chi-Square Tests
The Chi-Square Goodness-of-Fit Test
Page 14-01
[Navidi section 6.10; Devore chapter 14]
One simple test to determine whether or not a given random sample is consistent with some
probability distribution is to compare the numbers

ENGI 4421
Continuous Probability Distributions
Example 6.01:
Page 6-01
[Navidi section 2.4; Devore sections 4.1-4.2]
Exact lifetime is a continuous random quantity, but
Measured lifetime to the nearest minute is a discrete random quantity.
In a bar chart,

ENGI 4421
Simple Linear Regression
Page 15-01
Simple Linear Regression
[Navidi Sections 7.2-7.4; Devore Chapter 12]
[This topic is treated somewhat differently here from the approach in the textbooks.]
Sometimes an experiment is set up where the experimen

ENGI 4421 Probability and Statistics
Faculty of Engineering and Applied Science
Problem Set 1 Solutions
Descriptive Statistics
1.
If, in the set of values
cfw_ 11, 12, 13, 14, 15, 16, 17
an error causes the value 15 to be replaced by 150,
(a) what effect

ENGI 4421 Probability and Statistics
Faculty of Engineering and Applied Science
Problem Set 2 Solutions
[Introduction to Probability; odds, Venn diagrams, decision trees]
1.
Draw a decision tree, involving choice and chance forks, for each of
(a) a person

ENGI 4421 Probability and Statistics
Faculty of Engineering and Applied Science
Problem Set 4 Solutions
Conditional probability, Bayes' theorem
1.
One urn contains five red balls and five green balls. A second urn contains eight red
balls and one green ba

ENGI 4421 Probability and Statistics
Faculty of Engineering and Applied Science
Problem Set 5 Solutions
Probability distributions, expectation, variance
1.
A discrete function of x is defined by
1
8
p ( x) = 3
4
0
( x = 1, + 1)
( x = 0)
( otherwise )
(a)