En505 engineering statistics student notes

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  • 1. EN505 Engineering StatisticsFernando Tovia, Ph.D. 1 RANDOM VARIABLES AND PROBABILITY1.1 Random VariablesDefinition 1.1 A random experiment is an experiment such that the outcome cannot bepredicted in advance with absolute precision.Definition 1.2 The set of all possible outcomes of a random experiment is called thesample space. The sample space is denoted by . An element of the sample space isdenoted by .Example 1.1 Construct the sample space for each of the following random experiments:1. flip a coin2. toss a die3. flip a coin twiceDefinition 1.3 A subset of is called an event. Events are denoted by italicized, capitalletters.Example 1.2 Consider the random experiment consisting of tossing a die. Describe thefollowing events. 1. A = the event that 2 appears 2. B = the event that an even number appears 3. C = the event that an odd number appears 4. D = the event that a number appears 5. E = the event that no number appearsThe particular set that we are interested in depends on the problem being considered.However, a good thing to do when beginning any probability modeling problem is toclearly define all the events of interest.One graphical method of describing events defined on a sample space is the Venndiagram. The representation of an event using a Venn diagram is given in Figure 1.1 Notethat the rectangle corresponds to the sample space, and the shaded region corresponds tothe event of interest.Figure 1.1 Venn Diagram for Event A 1

2. Definition 1.4 Let A and B be two event defined on a sample space . A is a subset of B,denoted by A B, if an only if (iff), A, B. (Figure 1.2)Figure 1.2 Venn Diagram for A BDefinition 1.5 Let A be an event defined on a sample space . Ac iff A. Ac iscalled the complement of A. (Figure 1.3)Figure 1.3 Venn Diagram for AcDefinition 1.6 Let A and B be two events defined on the sample space . A B iff A or B (or both). A B is called the union of A and B (see Figure 1.4). Figure 1.4 Venn Diagram for A BLet {A1, A2, } be a collection of events defined on a sample space. U Aj j =1iff some j = 1, 2, AjUAj =1 j is called the union of {A1, A2, }Definition 1.7 Let A and B be two events defined on the sample space . A B iff A and B. A B is called the intersection of A and B (see Figure 1.5).Figure 1.5 Venn Diagram forLet {A1, A2, } be a collection of events defined on a sample space. I Aj j =1iff A j = 1, 2, IAj =1 j is called the intersection of {A1, A2, }Example 1.3 (example 1.2 continued) 1. Bc = C2 3. 2. B C = D3. A B = BTheorem 1.1 Properties of ComplementsLet A be an event defined on a sample space . Then(a)(b)Theorem 1.2 Properties of the UnionsLet A, B, C be events defined on a sample space . Then(a)(b)(c)(d)(e)Example Prove Theorem 1.2 (c)Theorem 1.3 Properties of the IntersectionLet A, B, and C be events defined on the sample space . Then(a)(b)(c)(d)(e)Example 1.6 Prove theorem 1.3 (b)3 4. Theorem 1.4 Distribution of Union and IntersectionLet A, B and C be events defined in the sample space . Then (a) A (B C) = (A B) (AC) (b) A (B C) = (A B) (A C)Theorem 1.5 DeMorgans LawLet A, B and C be events defined in the sample space . Then (a) (A B)c = Ac Bc (b) (A B) c = Ac BcDefinition 1.8 Let A and B be two events defined in the sample space . A and B are saidto be mutually exclusive or disjoint iff A B = (Figure 1.6). A collection of events{A1, A2, }, defined on a sample space , is said to be disjoint iff every pair of eventsin the collection is mutually exclusive. Figure 1.6 Venn Diagram for Mutually Exclusive EventsDefinition 1.9 A collection of events {A1, A2, , An} defined on a sample space , issaid to be a partition (Figure 1.7) of iff(a) the collection is disjoint n (b) UA j = j =1Figure 1.7 Venn Diagram for a PartitionExample 1.7 (Example 1.2 continued) Using the defined event, identify: (a) a set of mutually exclusive events (b) a partition of the sample space 4 5. Definition 1.10 A collection of events, F, defined on a sample space , is said to be afield iff (a) F, (b) if A F, then Ac F, thenn U A F, j =1jWe use fields to represent all the events that we are interested in study. To construct afield: 1. we start with 2. is inserted by implication (Definition 1.10 (b) 3. we then add the events of interest 4. we then add complements and unionsExample 1.8 Suppose we perform a random experiment which consists of observing thetype of shirt worn by the next person entering a room. Suppose we are interested in thefollowing events.L = the shirt that has long sleevesS = the shirt has short sleevesN = the shirt has no sleevesAssuming that {L, S, N} is a partition of , construct an appropriate fieldTheorem 1.6 Intersection are in FieldsLet F be a field of events defined in the sample space . Then if A1, A2, , An F,then nIA Fj =1 jExample 1.9 Prove that if A, B F, then A B F.5 6. Any meaningful expression containing events of interest, , , and c can be shown to be in the field. Definition 1.11 Consider a set of elements, such as S = {a, b, c}. A permutation of the elements is an ordered sequence of elements. The number of permutations of n different elements is n! where n! = n x (n-1) x (n-2) x x 2 x 1 Example 1.10 List all the permutations of the elements S Definition 1.12 The number of permutations of subsets of r elements selected from a set of n different elements is Another counting problem of interest is the number of subsets of r elements that can be selected from a set of n elements. Here the order is not important, and are called combinations. Definition 1.13 The number of combinations, subsets of size r that can be selected from a set of n elements, is denoted as Example 1.11 The EN505 class has 13 students. If teams of 2 students can be selected, how many different teams are possible?1.2 Probability Probability is used to quantify the likelihood, or chance, that an outcome of a random experiment will occur. Definition 1.14 A random variable is a real-valued function defined on a sample space. Random variables are typically denoted by italicized capital letters. Specific values taken on by a random variable are typically denoted by italicized, lower-case letters. Definition 1.15 A random variable that can take on a countable number of values is said to be a discrete random variable. Definition 1.16 A random variable that can take on an uncountable number of values is said to be a continuous random variable. 6 7. Definition 1.17 The set of possible values for a random variable is referred as a range ofa random variable.Example 1.12 Consider the following experiments of random variables, define therandom variable, identify the range for each random variable, and classify it as discrete orcontinuous. 1. flip a coin 2. toss a die until a 6 appears 3. quality inspection of a shipment of manufactured items. 4. arrival of customer to a bankDefinition 1.18 Let be the random space for some random experiment. For any eventdefined on , Pr() is a function which assigns a number to the event. Pr(A) is called theprobability of event A provided the following conditions hold:(a)(b)(c)Probability is used to quantify the likelihood, or chance, that an event will occur withinthe sample space. 7 8. Whenever a sample consists of N possible outcomes , the probability of each outcome is1/NTheorem 1.7 Probability Computational RulesLet A and B events defined on a sample space , and let {A1, A2, , An} be a collectionof events defined on . Then(a)(b)(c)(d)(e)(f)Corollary 1.1 Union of Three or More EventsLet A, B, C and D be events defined on a sample space . Then,Pr( A B C ) = Pr( A) + Pr( B ) + Pr(C ) Pr( A B ) Pr( A C ) Pr( B C ) + Pr( A B C )andPr( A B C D ) = Pr( A) + Pr( B) + Pr(C ) + Pr( D) Pr( A B) Pr( A C ) Pr( A D ) Pr( B C ) Pr( B D) Pr(C D) + Pr( A B C ) + Pr( A B D) + Pr( B C D) + Pr( A B C D )Example 1.11 Let A, B and C be events defined on a sample space Pr(A) = 0.30Pr(Bc) = 0.60Pr(C) = 0.20Pr(A B) = 0.50Pr( B C ) = 0.05A and C are mutually exclusiveCompute the following probabilities(a) Pr(B) 8 9. (b) Pr( B C ) = (c) Pr( A B ) (d) Pr( A C ) (e) Pr( A C ) (f) Pr( B C c ) (g) Pr( A B C ) =1.3 IndependenceTwo events are independent if any one of the following equivalent statements is true 9 10. (1) P(A|B) = P(A) (2) P(B|A) = P(B) (3) P ( A B ) = P ( A) P ( B ) Example 2.29 (book work in class)1.4 Conditional ProbabilityDefinition 1.19. Let A and B events define on a sample space B . We refer toPr(A|B) as the conditional probability of event given the occurrence of event B, wherePr( AC | B) =probability of not A given B 10 11. note that Pr(A|Bc) 1-Pr(A|B)Example 1.12A semiconductor manufacturing facility is controlled in a manner such that 2% ofmanufactured chips are subjected to high levels of contamination. If a chip is subjected tohigh levels of contamination, there is a 12% chance that it will fail testing. What is theprobability that a chip is subjected to high levels of contamination and fails upon tests?c=f=Pr(High c level) =Pr(Fail | high c level) =Pr(FC) =Example 1.13An air quality test is designed to detect the presence of two molecules (molecule 1 andmolecule 2). 17% of all samples contain both molecules, and 48% of all samples containmolecule 1. If a sample contains molecule 1, what is the probability that it also containsmolecule 2?M1 = molecule 1M2 = molecule 2Pr(M1M2) =Pr(M1) =Pr(M2|M1) =Theorem 1.8 Properties of Conditional ProbabilityLet A and B be non-empty event defined on a sample space . Then (a) If A and B are mutually exclusive then Pr(A|B) = 0 (b) If A B then Pr(A|B)>=Pr(A)11 12. (c) If B A then Pr(A|B) =1Theorem 1.9 Law of Total Probability Part 1Let A and B be events defined on a sample space A , B , Bc . ThenExample 1.15A certain machines performance can be characterized by the quality of a key component.94% of machines with a defective key component will fail. Whereas only 1% ofmachines wi