ENGI 4421
Example 3.01
Counting Techniques for Probability
[Navidi Section 2.2; Devore Section 2.3]
Three cards, labelled A , B and C , are in an urn.
In how many ways can three cards be drawn
(a) with replacement?
(b) without replacement?
(c) without rep
ENGI 4421
Dr. Reza Shahidi
March 26, 2015
Problem Set 8
Normal Distribution, One-sided Confidence Intervals
1. Suppose that 10% of all steel shafts produced by a certain process are
nonconforming but can be reworked (rather than having to be scrapped).
Co
ENGI 4421
Dr. Reza Shahidi
March 8, 2015
Problem Set 7 Solutions
Discrete Probability Distributions; Exponential Distribution
1.
Suppose at a certain university consisting of 1000 students, exactly 300 of
them own their own car. A random sample of 5 stude
ENGI 4421
Dr. Reza Shahidi
February 21, 2015
Problem Set 5 Solutions
Probability distributions, expectation, variance
1. A discrete function of x is defined by:
1
,(x = 1,+1)
6
2
p(x) =
,(x = 0)
3
0(otherwise)
(a) Verify that p(x) is a well-defined p
ENGI 4421
Dr. Reza Shahidi
March 29, 2015
Problem Set 10 Solutions
Chi-Square Tests, Simple Linear Regression
1. According to genetic theory, the cross of two cereal crops should produce
plants with red, yellow, or white seeds in the ratio 9:3:4. Does the
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
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
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
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
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
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
Example 3.01
Counting Techniques for Probability
[Navidi Section 2.2; Devore Section 2.3]
Three cards, labelled A , B and C , are in an urn.
In how many ways can three cards be drawn
(a) with replacement?
(b) without replacement?
(c) without rep
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
Propagation of Error
Page 8-01
[Navidi Chapter 3; not in Devore]
Propagation of Error
Any realistic measurement procedure contains error.
Any calculations based on that measurement will therefore also contain an error.
The errors are propagated
ENGI 4421
Dr. Reza Shahidi
March 29, 2015
Problem Set 9 Solutions
Hypothesis Tests
1. The sample average unrestrained compressive strength for 45 specimens
of a particular type of brick was computed to be 3107 psi, and the sample
standard deviation was 18
ENGI 4421
Dr. Reza Shahidi
January 26, 2015
Problem Set 2 Solutions
Introduction to Probability; odds, Venn diagrams, decision trees
1. Suppose you wish to buy a refurbished TV which comes with a
90-day warranty. You are given the option of purchasing a 1
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
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
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
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
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
Joint Probability Distributions
Joint Probability Distributions
Page 7-01
[Navidi sections 2.5 and 2.6; Devore sections 5.1-5.2]
The joint probability mass function of two discrete random quantities X, Y is
p(x, y) = P[ X = x and Y = y ]
The mar
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
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