1. Engineering Probability and Statistics
Dr. Leonore Findsen
Department of Statistics

2. Outline Sets and Operations Counting Sets Probability Random Variables
Standard Distribution Functions
Statistical Treatment of Data
Statistical Inference
Sets and Operations
Set Operations
Venn Diagrams
Product Sets
Counting Sets
Permutations
Combinations
Probability
Definitions
General Character of Probability
Complementary Probabilities
Joint Probability
Conditional Probability
Random Variables
Probability Density Functions
Properties of Probability Density Functions
Standard Distribution Functions
Binomial
Normal
Statistical Treatment of Data
Frequency distribution
Standard Statistical measures
Statistical Inference
t-Distribution
I will briefly cover all of the items in the outline; however, there is not enough time to cover all of the parts in detail. Therefore, I will be emphasizing the parts that are most confusing to the students. Note: I do not have access to the FE study guide so I am using this from my personal experience from teaching statistics and recommendations from other faculty on campus.

3. Sets and Operations Venn Diagrams A set is a collection of objects.
An element of the set is one of the objects.
The empty set, , contains no objects.
Venn Diagrams
5 mins Sets and Operations - sets of objects may be defined by listing or stating properties (terminology and definitions)
5 mins Set Operations - sets can be manipulated mathematically and follow these laws: identity, complement, commutative, associative, de Morgan’s (definitions)
5 mins Venn Diagrams – Venn diagrams present set relationships graphically
5 mins Product Sets - The set of all ordered pairings of the elements of two sets is their product set, analogous to a pair of Cartesian axes when one set is arranged horizontally and the other set vertically, and the product set is the matrix.
Most engineers do not have problems with this section. A set is just a collection of objects. An element is one of those sets, the empty set contains no objects, this is not the same as ‘0’ which is something

4. Set Operations Union, U, A or B or both Intersection, ∩, A and B, AB
Complement, Ac, everything but A.
5 mins Sets and Operations - sets of objects may be defined by listing or stating properties (terminology and definitions)
5 mins Set Operations - sets can be manipulated mathematically and follow these laws: identity, complement, commutative, associative, de Morgan’s (definitions)
5 mins Venn Diagrams – Venn diagrams present set relationships graphically
5 mins Product Sets - The set of all ordered pairings of the elements of two sets is their product set, analogous to a pair of Cartesian axes when one set is arranged horizontally and the other set vertically, and the product set is the matrix.
Most engineers do not have problems with this section. A set is just a collection of objects. An element is one of those sets, the empty set contains no objects, this is not the same as ‘0’ which is something

5. Set Operations/Product Sets
Set Operations (de Morgan’s Laws)
(A U B)c = Ac ∩ Bc (A ∩ B)C = Ac U Bc
Product Sets – Cartesian Product
The set of all ordered pairings of the elements of two sets.
Example: A = {1,2}, B = {3,4} A X B ={(1,3), (2,3), (1,4), (2,4)}
5 mins Sets and Operations - sets of objects may be defined by listing or stating properties (terminology and definitions)
5 mins Set Operations - sets can be manipulated mathematically and follow these laws: identity, complement, commutative, associative, de Morgan’s (definitions)
5 mins Venn Diagrams – Venn diagrams present set relationships graphically
5 mins Product Sets - The set of all ordered pairings of the elements of two sets is their product set, analogous to a pair of Cartesian axes when one set is arranged horizontally and the other set vertically, and the product set is the matrix.
Most engineers do not have problems with this section. A set is just a collection of objects. An element is one of those sets, the empty set contains no objects, this is not the same as ‘0’ which is something

6. Basic Set Theory ExampleF
F ∩ P ∩ GC
= 6 G
Copyright Kaplan AEC Education, 2008

7. Copyright Kaplan AEC Education, 2008
Solution
Copyright Kaplan AEC Education, 2008

8. Counting Sets Finding the number of possible outcomes.
Counting the number of possibilities
Ways of counting
Sampling with or without replacement
Ordered or unordered
Product Rule
Permutations
Combinations
Complicated
5 mins Combinatorics – Finding the number of possible outcomes in a probability calculation can be performed by counting the elements in the outcome set, or by multiplying the number of elements in each subset
Product Rule – ordered pairs
10 mins Permutations – the order of the choosing is important
10 mins Combinations – the order of the choosing is not important
With replacement: daily 3, daily 4
Without replacement: lotto, powerball, megamillions
Ordered: pres, vp, treasurer – chi epsion
Unordered: you are taking a field trip and 3 people can go

9. Product Rule Ordered Pairs with replacement Formula: n1  ∙∙∙  nm
Examples:
Number of ways that you can combine alphanumerics into a password.
Number of ways that you can combine different components into a circuit.
5 mins Combinatorics – Finding the number of possible outcomes in a probability calculation can be performed by counting the elements in the outcome set, or by multiplying the number of elements in each subset
Product Rule – ordered pairs
10 mins Permutations – the order of the choosing is important - The elements of a set may be put in various ordered arrangements; each is a permutation
10 mins Combinations – the order of the choosing is not important - subsets may be drawn from a larger set in a manner similar to permutations but without regard for order

10. Permutations An ordered subset without replacement Formula Examples:
Number of ways that you can combine alphanumerics into a password if you can not repeat any symbols.
Testing of fuses to see which one is good or bad.
Choosing officers in a club.
5 mins Combinatorics – Finding the number of possible outcomes in a probability calculation can be performed by counting the elements in the outcome set, or by multiplying the number of elements in each subset
Product Rule – ordered pairs
10 mins Permutations – the order of the choosing is important - The elements of a set may be put in various ordered arrangements; each is a permutation
10 mins Combinations – the order of the choosing is not important - subsets may be drawn from a larger set in a manner similar to permutations but without regard for order

11. Combinations – ordered with and without replacement
With replacement: each letter can be repeated.
# of airports =(26)(26)(26) = 17,576 airports
Without replacement: each letter can not be repeated
# of airports =(26)(25)(24) = 15,600 airports
Copyright Kaplan AEC Education, 2008

12. Combinations An unordered subset without replacement Formula Examples:
Choosing members of a club to see who will be going to a national conference.
Selecting 3 red cards from a deck of 52 cards.
5 mins Combinatorics – Finding the number of possible outcomes in a probability calculation can be performed by counting the elements in the outcome set, or by multiplying the number of elements in each subset
Product Rule – ordered pairs
10 mins Permutations – the order of the choosing is important - The elements of a set may be put in various ordered arrangements; each is a permutation
10 mins Combinations – the order of the choosing is not important - subsets may be drawn from a larger set in a manner similar to permutations but without regard for order

13. Combinations - Example
# of teams = (15)(12)(8)(5) = 7,200
# of teams =
Copyright Kaplan AEC Education, 2008

14. Complicated Counting
How many different ways can you get a full house?

15. Probability Definitions
The probability of an event is the ratio of the number of times that it occurs to the number of times that everything occurs
Probability
5 mins Definitions – The probability of an event is the ratio of occurrences of that event to the occurrences of all the other mutually exclusive and equally probable events
5 mins General Character of Probability
10 mins Complementary Probabilities
10 mins Joint Probability
10 mins Conditional Probability

16. Probability - Properties
P() = 0, P(everything) = 1
P(E) = 1 – P(Ec)
Example: Consider the following system of components connected in a series. Let E = the event that the system fails. What is P(E)?
P(E) = 1 – P(SSSSS)
Probability
5 mins Definitions
5 mins General Character of Probability: The probability of an event may range from 0 to 1
10 mins Complementary Probabilities: The sum of the probability of an event happening and the probability of it not happening is 1
10 mins Joint Probability
10 mins Conditional Probability 5 4 3 2 1

17. Joint Probability P(A U B) = P(A) + P(B) – P(A ∩ B)
P(A ∩ B) = P(A)P(B) if A and B are independent
Probability
5 mins Definitions
5 mins General Character of Probability
10 mins Complementary Probabilities
10 mins Joint Probability – The joint probability of an event happening or another event happening (but not both) is the sum of their individual probabilities minus the probability of both happening. The probability of two independent events both happening is the probability of their individual probabilities
10 mins Conditional Probability

18. Joint Probability - Example
2-54: Given the following odds: In favor of event A 2:1 In favor of event B 1:5 In favor of event A or event B or both 5:1 Find the probability of event AB occurring? P(A U B) = P(A) + P(B) – P(A ∩ B) P(A ∩ B) = 0

19. Joint Probability - Example
The probability that a defective part is generated from Machine A is 0.01; the probability that a defective part is generated from Machine B is 0.02, What is the probability that both machines have defective parts? P(A ∩ B) = P(A)P(B) = (0.01)(0.02) =

20. Conditional Probability
Conditional Probability Definition
General Multiplication
P(A ∩ B) = P(A|B)P(B) = P(A)P(B|A)
P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B)
Bayes’ Theorem
Probability
5 mins Definitions
5 mins General Character of Probability
10 mins Complementary Probabilities
10 mins Joint Probability
10 mins Conditional Probability: The conditional probability of an event given another event is the ratio of their joint probability under Rule 3 to the probability of the second event (Bayes’ theorem)

21. Copyright Kaplan AEC Education, 2008
Bayes’ Theorem 1
P(D) = 0.6 P(E) = 0.2 P(F) = 0.2
P(B|D) = 0.12 P(B|E) = 0.04 P(B|F) = 0.10
P(B) = P(D)P(B|D) + P(E)P(B|E) + P(F)P(B|F) = 0.10
Copyright Kaplan AEC Education, 2008

22. Copyright Kaplan AEC Education, 2008
Bayes’ Theorem 2
Given that the car has bad tires, what is the probability that it was rented from Agency E?
P(D) = 0.6 P(E) = 0.2 P(F) = 0.2
P(B|D) = 0.12 P(B|E) = 0.04 P(B|F) = 0.10
P(B) = P(D)P(B|D) + P(E)P(B|E) + P(F)P(B|F) = 0.10
Copyright Kaplan AEC Education, 2008

23. Random Variables Definition
A random variable is any rule that associates a number with each outcome in your total sample space.
A random variable is a function.
5 mins Random Variables – mapping numerical elements of the sample space to real number intervals on the x-axis allows use of standard mathematical tools to solve probability problems
10 mins Probability Density Functions
10 mins Properties of Probability Density Functions

24. Probability Density Functions
The area under a pdf curve for an interval is the probability that an event mapped into that interval will occur.
P(a  X  b)
5 mins Random Variables
10 mins Probability Density Functions – the area under a probability density function curve for a given interval equals the probability that an event mapping into that interval will occur
10 mins Properties of Probability Density Functions

25. Cumulative Distribution Functions
P(X  a) = F(a)
5 mins Random Variables
10 mins Probability Density Functions – the area under a probability density function curve for a given interval equals the probability that an event mapping into that interval will occur
10 mins Properties of Probability Density Functions

26. Properties of pdfs Percentiles Mean E(h(x)) Variance p = F(a)
σ2 = Var(X) = E[(X – μ)2] = E(X2) – [E(X)]2
5 mins Random Variables
10 mins Probability Density Functions
10 mins Properties of Probability Density Functions: The mean of a probability density function can be calculated for discrete or continuous r.v. and for discrete or continuous random variables of a function, including the function g(x) = (x – u)2, whose expected value is the variance.

27. Cumulative Distribution Function - Example
Copyright Kaplan AEC Education, 2008

28. Cumulative Distribution Function – Example (cont)
Copyright Kaplan AEC Education, 2008

29. Standard Distribution Functions
There are some standard distributions that are commonly used. These are determined from either from the experiment or from analysis of the data.
5 mins Standard Distribution Functions – probability density functions may be found based either on analysis of the actual experiment and outcomes, or else on application of a known function from an appropriately similar experiment
10 mins Binomial
10 mins Normal
10 mins t-Distribution

30. Binomial Distribution
Experimental Conditions – BInS
B: Each trial can have only two outcomes (binary).
I: The trials are independent.
n: Know the number of trials
S: The probability of success is constant.
Want to find the number of successes.
Formula:
5 mins Standard Distribution Functions
10 mins Binomial – The binomial probability density function applies when there are binary alternative outcomes, and is the product of the binomial coefficient of the given number of occurrences (r) from a given number of trials (n) and the joint probability of r events occuring and n-r events occuring
10 mins Normal
10 mins t-Distribution

31. Binomial Discrete Distributions - Example
Let X = number of cars out of five that get a green light.
X ~ B(n,p) = B(5,0.7)
P(X  3)
= 1 – P(X < 3) = 1 – P(X  2) = 1 – F(2) = 1 – =
Copyright Kaplan AEC Education, 2008

32. Other Discrete Distributions
Hypergeometric
Like binomial but without replacement
Poisson
Like a binomial but with very low probability of success
Negative Binomial
Like binomial but want to know how many trials until a certain number of successes.

33. Normal Distribution Function
Continuous
This is the most commonly occurring distribution.
Systematic errors
A large number of small values equally likely to be positive or negative
5 mins Standard Distribution Functions
10 mins Binomial
10 mins Normal – If the distribution of outcomes does not reflect systematic errors, and the remaining elementary errors are small, numerous, and equally liekly to be positive or negative, then it will approximate a normal distribution (figures, table, formulas)
10 mins t-Distribution

34. Normal Distribution Function (cont)
The parameters of the normal distribution are μ and σ
The normal distribution can not be integrated so we use z-tables which are for the standard normal with μ = 0, σ = 1. The z-tables contain the cdf, (z).
To convert our distribution to the standard normal,

35. Normal Distribution Function - Example
F(c*) = 0.01 ==> c* =
σ = 0.86 kN/sq. m
Copyright Kaplan AEC Education, 2008

36. Statistical Treatment of Data
Most people need to visualize the data to get a feel for what it looks like.
In addition, summarizing the data using numerical methods is also helpful in analyzing the results.
5 mins Statistical Treatment of Data – Large data sets are difficult to understand in their raw form; they are more useful when organized and analyzed
5 mins Frequency distribution
10 mins Standard Statistical measures

37. Frequency Distribution
Frequency table
Histogram
Example
100 married couples between 30 and 40 years of age are studied to see how many children each couple have. The table below is the frequency table of this data set.
5 mins Statistical Treatment of Data
5 mins Frequency distribution – the same data set can be presented as a frequency table, frequency bar graph, frequency line graph, or cumulative frequency line graph
10 mins Standard Statistical measures

38. Frequency Distribution – Example (cont)
Kids
# of Couples
Rel. Freq 1 11
0.11 2 22
0.22 3 30
0.30 4 5
0.01 6
0.00 7
100
1.00
Frequency Distribution – Example (cont)

39. Numerical Statistical Measures
Measures of the central value
Mean
Median
Mode
Measures of variability
Range
Variance (standard deviation)
Interquartile range
5 mins Statistical Treatment of Data
5 mins Frequency distribution
10 mins Standard Statistical measures – Various numerical quantities measure central values of data set, and the degree to which a data set is concentrated around or deviates from a central value

40. Measures of Dispersion
Copyright Kaplan AEC Education, 2008

41. Copyright Kaplan AEC Education, 2008
Solution
Copyright Kaplan AEC Education, 2008

42. Statistical Inference Confidence Intervals
t- Distribution
Used when the population distribution is normal but σ is unknown
Tables will have to be provided if necessary
Confidence Intervals for μ
General form:
point estimator  critical value SEestimator
10 mins t-Distribution

43. Copyright Kaplan AEC Education, 2008
Interval Estimates
Copyright Kaplan AEC Education, 2008

44. Copyright Kaplan AEC Education, 2008
Solution
Copyright Kaplan AEC Education, 2008

45. Copyright Kaplan AEC Education, 2008
Solution (continued)
Copyright Kaplan AEC Education, 2008

46. Copyright Kaplan AEC Education, 2008
Solution (continued)
Copyright Kaplan AEC Education, 2008

47. Statistical Inference Hypothesis Testing - Procedure
Hypotheses
Ho: null hypothesis,  = 0
HA: alternative hypothesis,   0, > 0, < 0
Test statistic
Decision
  0:P(|T|>ts),  > 0:P(T>ts),  < 0:P(T<ts)
10 mins t-Distribution

48. Statistical Inference Hypothesis Testing - Errors
calculated/true
Ho true
Ho false
fail to reject Ho
correct
Type II, β
reject Ho
Type I, α
10 mins t-Distribution

49. Conclusion Sets and Operations Counting Sets Probability
Random Variables
Standard Distribution Functions
Statistical Treatment of Data
Statistical Inference
Sets and Operations
Set Operations
Venn Diagrams
Product Sets
Counting Sets
Permutations
Combinations
Probability
Definitions
General Character of Probability
Complementary Probabilities
Joint Probability
Conditional Probability
Random Variables
Probability Density Functions
Properties of Probability Density Functions
Standard Distribution Functions
Binomial
Normal
Statistical Treatment of Data
Frequency distribution
Standard Statistical measures
Statistical Inference
t-Distribution
I will briefly cover all of the items in the outline; however, there is not enough time to cover all of the parts in detail. Therefore, I will be emphasizing the parts that are most confusing to the students. Note: I do not have access to the FE study guide so I am using this from my personal experience from teaching statistics and recommendations from other faculty on campus.