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

2. Outline Sets (terminology only) Counting (Combinatorics)
Probability (conditional probabilities)
Probability Distributions (discrete, continuous, weighted average)
Standard Distribution Functions (normal, binomial)
Descriptive Statistics (mean, mode, standard deviation)
Statistical Inference (sampling distributions)
Confidence Intervals
Hypothesis testing
Linear Regression (goodness of fit)

3. Sets 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

4. Set Operations Union, U, A or B or both Intersection, ∩, A and B, AB
Complement, A’ or AC or 𝐴 , everything but A.

5. Counting Finding the number of possible outcomes.
Counting the number of possibilities
Ways of counting
Sampling with or without replacement
Ordered or unordered
Product Sets
Permutations
Combinations
Complicated

6. Product Sets 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.
Example
A = {1,2}, B = {3,4}
A X B ={(1,3), (2,3), (1,4), (2,4)}

7. Product Sets - Problem
What is the total number of possible outcomes for rolling a 4 – sided die 6 times? 46 = 4096

8. 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.

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Counting (Example)
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
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10. Permutations: Example
A jeweler has nine different beads and a bracelet design that requires four beads. How many different bracelets are there if it is linear?
P(9,4) = 9 x 8 x 7 x 6 = 3024
b) If the bracelet is a closed circle, there is no discernible difference when it is rotated. How many different bracelets are there?
Pring(9,4)= 𝑃(9,4) 4 = =756

11. Permutations: Example
How many ways can the letters of the word INDIANA be arranged? 𝑃 7;2,2,1,2 = 7! 2!2!1!2! =630

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.

13. Combinations - Example
# of teams = (15)(12)(8)(5) = 7,200
# of teams =
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14. Complicated Counting
1) How many different ways can you get a full house? 2) The probability that the first four cards dealt from the deck are A, A, A, 5 is 𝑃 𝑐𝑎𝑟𝑑𝑠 = 4 52 ∙ 3 51 ∙ 2 50 ∙ 4 49 =1.48× 10 −5

15. Probability Definition – General Character of Probability
The probability of an event is the ratio of the number of times that it occurs to the number of times that everything occurs

16. Probability - Properties
P() = 0, P(everything) = 1
P(E) = 1 – P(E’)
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) 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) – P(A,B)
P(A ∩ B) = P(A)P(B) if A and B are independent
(not listed in the reference handbook)

18. Joint Probability - Example
After Purdue wins a home game, the probability of a person having a car accident is The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is What is the probability of a person driving while intoxicated or having a car accident? P(I U A) = P(I) + P(A) – P(I ∩ A) = – 0.15 = 0.26

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. Complementary Probability - Example
Consider the following system of components connected in a series. Assuming that the failure of each of the components is independent of the other components and the probability that one of the component fails is 0.1, what is the probability that the whole circuit fails?
P(F) = 1 – P(S) = 1 – (1 – 0.1)5 = 0.409 5 4 3 2 1

21. Conditional Probability
Conditional Probability Definition
General Multiplication
P(A ∩ B) = P(A|B)P(B) = P(A)P(B|A) (given)
P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B)
Bayes’ Theorem (given)

22. Conditional Probability
The probability that both stages of a two-stage missile will function correction is The probability that the first stage will function correctly is What is the probability that the second stage will function correctly given that the first one does? 𝑃 2 1 = 𝑃(1,2) 𝑃(1) = =0.969

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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
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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
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25. Probability Distributions
Discrete
Population mean (weighted average)
𝜇= 𝑥∙ 𝑓𝑟𝑒𝑞𝑒𝑛𝑐𝑦(𝑥) 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟

26. Weighted Average: Problem
The scores for an exam are as follows: What is the expected value (weighted average) for a score on this exam? 𝜇=75∙ ∙ ∙ 2 25 =80.6
Frequency 13 10 2
Score 75 85 95

27. Probability Density Functions - continuous
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)

28. Cumulative Distribution Functions
P(X  a) = F(a)

29. Properties of pdfs Percentiles Mean E(g(x)) Variance p = F(a)
σ2 = Var(X) = E[(X – μ)2] = E(X2) – [E(X)]2

30. Cumulative Distribution Function - Example
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31. Cumulative Distribution Function – Example (cont)
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32. 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.

33. 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 X = the number of successes.
Formula:

34. Binomial Discrete Distributions - Example
The traffic light at State St. and River Road is either green, red or yellow. The following probabilities are for the Main St. traffic
P(green) = P(red) = 0.25 P(yellow) = 0.05
Out of the next 5 cars, what is the probability that exactly 1 car gets a green light?
Let X = number of cars out of five that get a green light.
X ~ B(n,p) = B(5,0.7)
𝑃 𝑋=1 = =
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35. Binomial Discrete Distributions - Example
The traffic light at State St. and River Road is either green, red or yellow. The following probabilities are for the Main St. traffic
P(green) = 0.7 P(red) = 0.25 Pr(yellow) = 0.05
b) Out of the next 5 cars, what is the probability that one or more cars get green lights?
Let X = number of cars out of five that get a green light.
X ~ B(n,p) = B(5,0.7)
𝑃 𝑋≥1 =1−𝑃 𝑋<1 =1−𝑃 𝑋=0
=1− =1− =
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36. 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.
Multinomial
Like binomial, but more than 2 options

37. 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

38. 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 tables contain a number of different functions.
F(-z) = 1 – F(z)
To convert the given distribution to the standard normal,

39. Normal Distribution Function - Example
Scores in a particular game have a normal distribution with a mean of 30 and a standard deviation of 5. Contestants must score more than 26 to qualify for the finals. The probability of being disqualified in the qualifying round is: 𝑃 𝑋<26 =𝑃 𝑍< 26−30 5 =𝑃 𝑍<0.8 =0.7781

40. Normal Distribution Function - Example
F(c*) = 0.01 ==> c* =
σ = 0.86 kN/sq. m
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41. Other Continuous Distributions
Exponential
Used in lifetimes and growth
Weibull
Use in lifetimes
Uniform
Equally likely situation.

42. Descriptive Statistics
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.

43. 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.

44. 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)

45. Numerical Statistical Measures
Measures of the central value
Mean
Median
Mode
Measures of variability
Range
Variance (standard deviation)
Interquartile range
Coefficient of Variation

46. Measures of Dispersion
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Solution
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48. Statistical Inference
Statistical inference is used to infer information from your data to the whole population.
Confidence Intervals
Hypothesis Testing
Linear Regression

49. Confidence Intervals t- Distribution Confidence Intervals for μ
Used when the population distribution is normal but σ is unknown
Tables are in the handbook
Confidence Intervals for μ
General form:
point estimator  critical value Seestimator
Sample size
𝑛= 𝑡 𝛼 2 ,𝑛−1 ∙𝑠 𝑒 2

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Interval Estimates
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Solution
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Solution (continued)
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53. Copyright Kaplan AEC Education, 2008
Solution (continued)
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54. Hypothesis Testing Hypotheses Ho: null hypothesis,  = 0
HA: alternative hypothesis,   0, > 0, < 0
Test statistic (there are more in the Handbook)

55. Hypothesis Testing (cont)
Decision Rule
  0: P(|T|>ts) ts ≥ tc or ts ≤ -tc
 > 0: P(T>ts) ts ≥ tc
 < 0: P(T<ts) ts ≤ tc
Decision
Reject H0: P-value ≤  or ts in rejection region above
Fail to reject H0: P-value >  or ts is not in the rejection region above.

56. Statistical Inference Hypothesis Testing - Errors
calculated/true
Ho true
Ho false
fail to reject Ho
correct
Type II, β
reject Ho
Type I, α
Example: Justice system
: Person is innocent and verdict is guilty
: Person is guilty and verdict is not guilty

57. Conclusion
Good Luck!