Engineering Probability and Statistics Dr. Leonore Findsen Department of Statistics

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)

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

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

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

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)}

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

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.

Copyright Kaplan AEC Education, 2008 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 Copyright Kaplan AEC Education, 2008

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 = 3024 4 =756

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

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.

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

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

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

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

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)

Joint Probability - Example After Purdue wins a home game, the probability of a person having a car accident is 0.09. The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is 0.15. 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) = 0.09 + 0.32 – 0.15 = 0.26

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) = 0.0002

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

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)

Conditional Probability The probability that both stages of a two-stage missile will function correction is 0.95. The probability that the first stage will function correctly is 0.98. 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) = 0.95 0.98 =0.969

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

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

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

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∙ 13 25 +85∙ 10 25 +95∙ 2 25 =80.6 Frequency 13 10 2 Score 75 85 95

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)

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

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

Cumulative Distribution Function - Example Copyright Kaplan AEC Education, 2008

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

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.

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:

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 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 = 5 1 0.7 1 0.3 4 =0.02835 Copyright Kaplan AEC Education, 2008

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− 5 0 0.7 0 0.3 5 =1−0.00243=0.99757 Copyright Kaplan AEC Education, 2008

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

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

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,

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

Normal Distribution Function - Example F(c*) = 0.01 ==> c* = -2.327 σ = 0.86 kN/sq. m Copyright Kaplan AEC Education, 2008

Other Continuous Distributions Exponential Used in lifetimes and growth Weibull Use in lifetimes Uniform Equally likely situation.

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.

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.

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)

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

Measures of Dispersion Copyright Kaplan AEC Education, 2008

Copyright Kaplan AEC Education, 2008 Solution Copyright Kaplan AEC Education, 2008

Statistical Inference Statistical inference is used to infer information from your data to the whole population. Confidence Intervals Hypothesis Testing Linear Regression

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

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

Copyright Kaplan AEC Education, 2008 Solution Copyright Kaplan AEC Education, 2008

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

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

Hypothesis Testing Hypotheses Ho: null hypothesis,  = 0 HA: alternative hypothesis,   0, > 0, < 0 Test statistic (there are more in the Handbook)

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.

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

Conclusion Good Luck!