Engineering Probability and Statistics Dr. Leonore Findsen Department of Statistics

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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.

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. 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

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

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 – 0.1631 = 0.8369 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.

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

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,

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

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

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

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 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

Measures of Dispersion Copyright Kaplan AEC Education, 2008

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

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

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

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

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

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.