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 – 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 c, everything but A.

Set Operations/Product Sets Set Operations (de Morgan’s Laws) – (A U B) c = A c ∩ B c (A ∩ B) C = A c U B c 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 =

Basic Set Theory Example P F G

Counting Sets Finding the number of possible outcomes. Counting the number of possibilities Ways of counting – Sampling with and without replacement – Product Rule – Permutations – Combinations – Complicated

Product Rule Ordered Pairs with replacement Formula: n 1  ∙∙∙  n m Examples: – Number of ways that you can combine alphanumerics into a password. – Number of ways that you can combine different components into a circuit.

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 of officers in a club.

Permutations – with and without replacement With replacement: each letter can be repeated. # of airports = Without replacement: each letter can not be repeated # of airports =

Combinations An unordered subset without replacement Formula Examples: – Choosing of officers in a club if one person can hold more than one office. – Selecting 3 red cards from a deck of 52 cards.

Combinations - Example a)# of teams = b)# of teams =

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 - Properties 0  P(E)  1 – P(  ) = 0, P(everything) = 1 P(E c ) = 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) 54321

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

Joint Probability - Example 2-54: Given the following odds: In favor of event A2:1 In favor of event B1:5 In favor of event A or event B or both5:1 Find the probability of event AB occurring?

Joint Probability - Example Let A = draw a diamond, B = draw a 4, P(4D)?

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

Bayes’ Rule 1

Bayes’ Rule 2 Given that the car has bad tires, what is the probability that it was rented from Agency E?

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.

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)

Cumulative Distribution Functions P(X  a)

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

Cumulative Distribution Function - Example a) b)

Cumulative Distribution Function – Example (cont) c)

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 1.Know the number of trials 2.Each trial can have only two outcomes. 3.The trials are independent. 4.The probability of success is constant. Formula:

Binomial Discrete Distributions - Example

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 my 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 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 z-tables contain the cdf,  (z). To convert our distribution to the standard normal,

Normal Distribution Function - Example

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.

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 CouplesRel. Freq

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

Measures of Dispersion

Statistical Inference 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

Interval Estimates

Statistical Inference Hypothesis Testing – H o : null hypothesis – H A : alternative hypothesis Errors calculated/trueH o trueH o false fail to reject H o correctType II = β reject H o Type I = αcorrect