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An introduction of CEE 3770 course.
Statistics& Applications
Mokhtarian, Patricia
Class Notes
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This 63 page Class Notes was uploaded by Aatmay Talati on Thursday August 25, 2016. The Class Notes belongs to CEE 3770 A at Georgia Institute of Technology - Main Campus taught by Mokhtarian, Patricia in Fall 2016. Since its upload, it has received 46 views. For similar materials see Statistics& Applications in Civil and Environmental Engineering (CEE) at Georgia Institute of Technology - Main Campus.

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Date Created: 08/25/16
CEE 3770 Introduction INTRODUCTION– Reading Assignment: Chapter 1; Sections 6.1, 6.3 The science of statistics is concerned with: 1. The descriptionof data sets; 2. Inferences about the parameters characterizing the system, "population", phenomenon, or process of concern; and 3. Testingof hypotheses using the data. Example: DDT Analyses on Fish Samples, Tennessee River, Alabama (Mendenhall and Sincich) The data set consists of 144 cases, 5 variables per case: location, species of fish, length, weight, and DDT level. How can we summarize or otherwise describe the 720 values in this data set? Notice that two variables (location and species) are qualitative, while the other three are quantitative. For the qualitative variables, we might tabulate the number of observations in each category and construct a frequency or relative frequency histogram. For the quantitative variables, we might take the "average" of each variable. We might take the average within each species category, and/or by location. We may develop a frequency distribution. If we had data over several points in time, we could try to determine any trends. We might wish to test if X is greater than Y (hypothesis testing). What might be a reasonable "X" and "Y" in this example? Further, we may try to find out if there's any relationship between the variables, e.g., Y = a + bX (modeling). What might be a reasonable model in this example? What does this data set represent? ... a finite number of measurements, on a limited number of species, at a finite number of locations, on selected days. The data set does not completely describe the state of every single fish in the entire Tennessee River for all time. That is, it's a sample, i.e., a subset of the measurements describing the DDT levels in fish of the Tennessee River. The complete set of measurements, or data that define the phenomenon of our concern, is called thepopulation. Often the population is simply a conceptual construct (M & R, p. 199). E.g., if “the time it takes to run a particularcomputer program”is of interest, the population might be defined as “the run time for all possible runs of that program”. "Sometimes the words 'population' and 'sample' are used to represent the objects upon which the measurements are taken" [rather than the measurements themselves] (M & S, p. 2). The meaning should be clear from the context. Int-1 CEE 3770 Introduction Describing these data We may do so using, for example, averages (a measure of central tendency) and standard devtiions (a measure of spread). What do these numbers stand for beyond being summary statistics of the data set at hand? Statistic: "a numerical descriptive measure computed from sample data" (Mendenhall and Sincich, 1995, p. 39). We believe these numbers say something about the "real" state of fish in the Tennessee River. That is, we believe we can infer the characteristics of the entire population from the statistics of this sample– with some level of confidence. Measures of central tendency Suppose you have a sample of N measurements, Y 1, ...NY . There are three common "measures of central tendency": the (arithmetic) mean, median, and mode. MEAN: N Y = ∑(Y )i/ N i=1 MEDIAN: The middle number when all measurements are arranged in ascending or des cending order. If there are an even number of measurements, the median is the mean of the middle two. MODE: The value that occurs with the greatest frequency. If no value occurs more than once, all values are the mode. • The mean and the median can be calculated only for quantitative data; the mode can apply to either quantitative or qualitative data. • In general, mean≠ median≠ mode. • The median is less affected by outliers, or skewness in the data, than is the mean. There- fore, often the median best describes a sample. • The mode is most useful for discrete (especially qualitative or categorical) data, where − the number of values the variable of interest can take on is small relative to the number of data points, and − the relative frequency of occurrence is of interest (e.g., a supplier of carpenter's materials selling 3 lengths of nails) • Beware of bi -modal distributions. In this case, the mean can fall into a "trough" and represent very few observations. Int-2 CEE 3770 Introduction • We most often use the mean, because we can do more with it matehmatically. But be aware of its sensitivity to outliers, and the possibility of bi -modal distributions (the two cases in which the mean may not be a useful measure of central tendency.)That's whyit's important to get a sense of the actual distribution of the data. One way of doing that is to look at measures of variation, or spread. Measures of spread RANGE: largest measurement -smallest measurement • Can suggest existence of outliers, but is a relatively insensitive measure (two data sets can have identical ranges but very different distributions). VARIANCE: N (∑ Yi)2 N 2 N 2 i=1 N 2 2 ∑ (Yi-Y ) ∑ Yi - ∑ Y i - NY s2= i=1 = i=1 N = i=1 N -1 N -1 N -1 STANDARD DEVIATION: s = √s 2 • The variance and standard deviation answer the question: On averge, how different are the data points from the mean? The more spread out they are, the bigger (in magnitudY −Y i is, therefore the bigger the variance is –and vice versa. In the extreme, ii alY , then s2= 0. Empirical Rule If the data have a "mound-shaped" frequency distribution, then about • 68% of the data points will lie within 1 standard deviation either side of the mean; • 95% will lie within 2 standard deviations; and • nearlyall will lie within 3 standard deviations of the mean. Int-3 CEE 3770 Introduction DDT Analyses on Fish Samples, Tennessee River, Alabama (Mendenhall and Sincich, 4th ed., 1995) Location Species Length Weight DDT Location Species Length Weight DDT (cm) (gm) (ppm) (cm) (gm) (ppm) 1 FCM5 channel catfish 42.5 732 10.00 73 TRM300 small mouth buffalo 35.5 1300 1.30 2 FCM5 channel catfish 44.0 795 16.00 74 TRM300 small mouth buffalo 46.0 1365 4.80 3 FCM5 channel catfish 41.5 547 23.00 75 TRM300 small mouth buffalo 45.0 1437 5.10 4 FCM5 channel catfish 39.0 465 21.00 76 TRM300 small mouth buffalo 44.5 1460 5.10 5 FCM5 channel catfish 50.5 1252 50.00 77 TRM300 small mouth buffalo 49.0 1671 4.00 6 FCM5 channel catfish 52.0 1255 150.00 78 TRM300 small mouth buffalo 47.5 1717 10.00 7 LCM3 channel catfish 40.5 741 28.00 79 TRM305 channel catfish 35.0 613 12.00 8 LCM3 channel catfish 48.0 1151 7.70 80 TRM305 channel catfish 51.0 353 22.00 9 LCM3 channel catfish 48.0 1186 2.00 81 TRM305 channel catfish 42.5 909 10.00 10 LCM3 channel catfish 43.5 754 19.00 82 TRM305 channel catfish 38.0 886 11.00 11 LCM3 channel catfish 40.5 679 16.00 83 TRM305 channel catfish 41.0 890 17.00 12 LCM3 channel catfish 47.5 985 5.40 84 TRM305 channel catfish 47.0 1031 9.70 13 SCM1 channel catfish 44.5 1133 2.60 85 TRM310 channel catfish 45.0 1083 12.00 14 SCM1 channel catfish 46.0 1139 3.10 86 TRM310 channel catfish 45.5 864 4.70 15 SCM1 channel catfish 48.0 1186 3.50 87 TRM310 channel catfish 45.0 886 6.00 16 SCM1 channel catfish 45.0 984 9.10 88 TRM310 channel catfish 45.0 965 3.80 17 SCM1 channel catfish 43.0 965 7.80 89 TRM310 channel catfish 39.0 537 17.00 18 SCM1 channel catfish 45.0 1084 4.10 90 TRM310 channel catfish 40.5 630 12.00 19 TRM275 channel catfish 48.0 986 8.40 91 TRM310 small mouth buffalo 46.0 1486 1.40 20 TRM275 channel catfish 45.0 1023 15.00 92 TRM310 small mouth buffalo 47.0 1743 6.10 21 TRM275 channel catfish 49.0 1266 25.00 93 TRM310 small mouth buffalo 48.5 2061 2.80 22 TRM275 channel catfish 50.0 1086 5.60 94 TRM310 small mouth buffalo 48.0 1707 4.80 23 TRM275 channel catfish 46.0 1044 4.60 95 TRM310 small mouth buffalo 38.0 862 5.70 24 TRM275 channel catfish 52.0 1770 8.20 96 TRM310 small mouth buffalo 38.5 911 3.30 25 TRM280 channel catfish 48.0 1048 6.10 97 TRM315 channel catfish 29.5 476 3.30 26 TRM280 channel catfish 51.0 1641 13.00 98 TRM315 channel catfish 42.0 743 3.70 27 TRM280 channel catfish 48.5 1331 6.00 99 TRM315 channel catfish 47.5 1128 9.90 28 TRM280 channel catfish 51.0 1728 6.60 100 TRM315 channel catfish 43.5 848 6.80 29 TRM280 channel catfish 44.0 917 5.50 101 TRM315 channel catfish 47.5 1091 13.00 30 TRM280 channel catfish 51.0 1398 11.00 102 TRM315 channel catfish 43.5 715 8.80 31 TRM280 small mouth buffalo 49.0 1763 4.50 103 TRM320 channel catfish 47.5 983 57.00 32 TRM280 small mouth buffalo 46.0 1459 4.20 104 TRM320 channel catfish 51.5 1251 96.00 33 TRM280 small mouth buffalo 52.0 2302 3.00 105 TRM320 channel catfish 49.5 1255 360.00 34 TRM280 small mouth buffalo 46.0 1614 2.30 106 TRM320 channel catfish 47.0 1152 130.00 35 TRM280 small mouth buffalo 46.0 1444 2.50 107 TRM320 channel catfish 47.5 1085 13.00 36 TRM280 small mouth buffalo 48.0 2006 6.80 108 TRM320 channel catfish 47.0 1118 61.00 37 TRM285 channel catfish 44.0 936 19.00 109 TRM320 small mouth buffalo 36.0 1285 12.00 38 TRM285 channel catfish 42.0 1058 7.20 110 TRM320 small mouth buffalo 34.5 1178 33.00 39 TRM285 channel catfish 42.5 800 6.00 111 TRM320 small mouth buffalo 44.5 1492 48.00 40 TRM285 channel catfish 45.5 1087 10.00 112 TRM320 small mouth buffalo 46.0 1524 10.00 41 TRM285 channel catfish 48.0 1329 12.00 113 TRM320 small mouth buffalo 46.0 1473 44.00 42 TRM285 channel catfish 44.0 897 2.80 114 TRM320 small mouth buffalo 32.5 520 0.43 43 TRM285 large mouth bass 28.5 778 0.48 115 TRM325 channel catfish 46.0 863 1100.00 44 TRM285 large mouth bass 26.0 532 0.18 116 TRM325 channel catfish 40.0 549 9.40 45 TRM285 large mouth bass 25.5 441 0.34 117 TRM325 channel catfish 43.5 810 4.10 46 TRM285 large mouth bass 25.0 544 0.11 118 TRM325 channel catfish 46.5 908 2.80 47 TRM285 large mouth bass 23.0 393 0.22 119 TRM325 channel catfish 43.0 804 0.74 48 TRM285 large mouth bass 28.0 733 0.80 120 TRM325 channel catfish 47.5 1179 14.00 49 TRM290 channel catfish 41.0 961 8.70 121 TRM330 channel catfish 32.0 556 22.00 50 TRM290 channel catfish 44.0 886 22.00 122 TRM330 channel catfish 40.5 659 9.10 Int-4 CEE 3770 Introduction 51 TRM290 channel catfish 41.0 678 13.00 123 TRM330 channel catfish 51.5 1229 140.00 52 TRM290 channel catfish 42.0 1011 3.50 124 TRM330 channel catfish 48.0 1050 4.20 53 TRM290 channel catfish 42.5 947 9.30 125 TRM330 channel catfish 47.0 952 12.00 54 TRM290 channel catfish 44.0 989 21.00 126 TRM330 channel catfish 41.0 826 2.00 55 TRM290 small mouth buffalo 43.5 1291 3.40 127 TRM330 small mouth buffalo 33.5 599 0.30 56 TRM290 small mouth buffalo 46.5 1186 13.00 128 TRM330 small mouth buffalo 47.0 1704 1.20 57 TRM290 small mouth buffalo 43.0 1293 5.60 129 TRM340 channel catfish 50.0 1207 7.10 58 TRM290 small mouth buffalo 47.0 1709 12.00 130 TRM340 channel catfish 45.0 911 180.00 59 TRM290 small mouth buffalo 46.0 1425 21.00 131 TRM340 channel catfish 49.0 1498 1.50 60 TRM290 small mouth buffalo 41.0 1176 8.00 132 TRM340 channel catfish 49.5 1496 2.40 61 TRM295 channel catfish 36.0 980 12.00 133 TRM340 channel catfish 50.0 1142 4.30 62 TRM295 channel catfish 47.5 1176 6.00 134 TRM340 channel catfish 45.0 879 3.90 63 TRM295 channel catfish 41.5 989 4.70 135 TRM340 small mouth buffalo 32.5 525 0.99 64 TRM295 channel catfish 49.5 1084 31.00 136 TRM340 small mouth buffalo 38.0 806 0.45 65 TRM295 channel catfish 46.0 1115 5.20 137 TRM340 small mouth buffalo 38.5 694 2.50 66 TRM295 channel catfish 46.5 724 27.00 138 TRM340 small mouth buffalo 36.0 643 0.25 67 TRM300 channel catfish 36.0 847 18.00 139 TRM345 large mouth bass 26.5 514 0.58 68 TRM300 channel catfish 37.0 876 7.50 140 TRM345 large mouth bass 23.5 358 2.00 69 TRM300 channel catfish 35.0 844 3.00 141 TRM345 large mouth bass 30.0 856 2.20 70 TRM300 channel catfish 36.0 908 13.00 142 TRM345 large mouth bass 29.0 793 7.40 71 TRM300 channel catfish 48.0 1358 7.30 143 TRM345 large mouth bass 17.5 173 0.35 72 TRM300 channel catfish 49.0 1019 15.00 144 TRM345 large mouth bass 36.0 1433 1.90 Int-5 CEE 3770 Probability For the next few weeks, we will be studying basic principles of probability. How does that relate to the statistical ideas we presented in the introduction? In probability, we draw conclusions about a sample based on knowledge (or assumptions) about the population. In statistics, we draw conclu- sions about the population based on knowledge about a sample. More accurately, in statistics we compare knowledge about the sample (e.g. the sample mean, variance, and other des criptive statistics) against assumptions or hypotheses about the population. We see how compatible our sample knowledge is with our population assumptions, by evaluating the probability of observing the sample characteristics that we do, under our assumptions about the population. If our assump- tions about the population yield a very small probability of observing the sample outcomes that in fact occur, it causes us to re-evaluate our assumptions about the population– i.e. to reject our ori- ginal hypotheses and consider new ones . RANDOM EXPERIMENTS AND SAMPLE SPACES– Reading: Section 2.1 RandomExperiments: A random experimentis a procedure whose outcome cannot be predetermined. The outcome is observed in a specific way, e.g., - throw three dice and observethe sum of the face-up numbers, or - throw three dice and observe the largest face-p number. The description of the experiment includes the observation to be made; the same action (e.g., throwing three dice) can involve different observations, or experiments. A possible outcome of an experiment that cannot be decomposed is called a sample point , or simple event. For example, consider the experiment: Toss a true coin and observe the side that turns up. This experiment yields two sample points: Head (H), and Tail (T). Note the idealization involved in the above definition of the set of sample points; the possibility that the coin lands on its edge, etc., is ignored (see Gore’s comment on the 2000 Presidential election in the Quotes file). Sample Space: The set of all sample points, or all possible outcomes, of an experiment is called a sample space. In the above example, the sample space is {H, T}. Prb-1 CEE 3770 Probability Examples: Several examples of experiments and sample spaces are presented below. O bserve that these sample spaces can be classified into several types. As noted above, the outcome of an experiment is observed (or measured) in a specific way and different ways of observing outcomes define different experiments because they lead to di fferent sample spaces. Therefore a sample space is not fixed by the action of errming the experiment by itself but by the point of view adopted by the observer. See Examples 1a-c and 7a, c below. In the following discussion, we will denote an experiment by E, and a sample space by S. Example 1a: E: Toss a true coin three times and observe the sequence of heads and tails. S: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Example 1b: E: Toss a true coin three times and record the number of heads. S: {0, 1, 2, 3} Example 1c: E: Toss a true coin three times and observe how many times it changes from a head to a tail or a tail to a head. S: {0, 1, 2}. Exercise 1: Enumerate the experimental outcomes for each sample point in the sample spaces of Experiments 1a-1c. Example2: E: A "lot", or batch, composed of N motors contains D defective items (D is known and 0 < D ≤ N). The motors are drawn from the lot one by one without replacement until the last defective item is found. Observe how many draws are made. S: {D, D+1, ..., N} Example 3a: E: Consider the same lot as in Example 2. The motors are drawn from the lot one by one three times without replacement. Record the number of defective motors among the three drawn. S: {max(0, 3-[N-D]), ..., min(3,D)}, where “max(0, 3 -[N-D])” refers to the larger of 0 and 3-[N-D], and “min(3,D)” refers to the smaller of 3 and D. Example 3b: E: Consider the same lot as in Example 2. The motors are drawn from the lot one by one three times with replacement (i.e., a motor is drawn, then Prb-2 CEE 3770 Probability returned back to the lot before the next draw). Record the number of defective motors among the three drawn. S: {0, 1, ..., 3} Example 4: E: In a module for a digital computer circuit, a diode is pulse welded between two strips of nickel ribbon. Each weld is visually inspected and probed at each end before classification as good, G, or defective, D. S: {GG, GD, DG, DD} All these experiments have finite sample space.s Example 5: E: Count the number of hits on a certain websitein a 24-hour period. S: {0, 1, 2, ... } The sample space in this example consists of all non-negative integers and is said to be countably infinite. In both cases, there are a finite number of sample points within any bounded segment of the real line (technically, though, the set of all rationalnumbers is also considered to be countably infinite, eventhough there are an infinite number of them in any bounded segment). Finite sample spaces and countably infinite sample spaces are both count able. They are also called discrete sample spaces. Example 6: E: A light bulb is manufactured, put on life tes t, and aged to failure. The elapsed time (in hours) at failure iscorded. S: {t: t ≥0} = R+ This sample space, consisting of all non- negative real numbers, is noncountable or nondenumerable. There are an infinite number of sample points within any bounded segment of the real line. It is also called a continuous sample spac.e Example 7a: E: Count the number of cars that pass a fixed reference point on a roadway during an interval of length T (> 0), where T is determined before the experiment. S: {0, 1, 2, ...}. Example 7b: E: Measure the elapsed time between the arrivals of one car and the next at a fixed reference point. + S: {R } Prb-3 CEE 3770 Probability Example 7c: E: Observe the arrival of cars at an intersection by measuring the elapsed time betweeneachsuccessive arrival. + + + S: {R 1 R ,2R ,3...} EVENTS Definition: An eventis a subset of a sample space, or a set of sample points. Examples: Consider the coin tossing experiment of Example 1a. We can define events such as: A: All the coin tosses give the same face. A': Head appears exactlytwice. In the first case, event A consists of sample points HHH and TTT, or A = {HHH, TTT}. In the second case, A' = {HHT, HTH, THH}. Events andSet Theory: Since an event is a set, the intersection, union, and complement operations are defined for events, and the laws and properties of set theory hold. Several basic concepts in set theory are summarized here for your convenience. Set: Aggregate or collection of objects (elements). Examples: A = {5, 6, 7, 8} A = {x: x ∈R, 0 ≤x ≤ 1} where R is the set of all real numbers, and the symbol∈" indicates that element x belongs to set R. We will use an upper-case letter to refer to a set, and a lowe-rcase letter to refer to an element. Space(S): The set of all elements. Empty Set (∅): A set with no element. Subset: A set whose elements are all also elements of another set. Superset: A set completely containing another set. The following expression is used to indicate that A is a subsetof B: A ⊆ B. If A ⊆ B, but A =/ B, then we say A is a proper subsetof B, and write A ⊂B. To indicate that B is a supersetof A, we can write B ⊇A. Prb-4 CEE 3770 Probability Venn Diagrams: Venn diagrams are a useful tool in expressing relationships among sets and subsets. Complement: Thecomplement of set A (A) is c c A = {x: x ∈S, x ∉A}, which is the set of all elements in S that are not members of" A"). Intersection: Theintersectionof set A and B is∈ A ∩ B = {x: x ∈A and x ∈ B}. c A ∩ B = {x: x ∈ A and x ∉B} c = {x: x ∈A and x ∈B } is denoted A \B. Union: Theunion of A and B is A B = {x: x ∈A or x ∈ B}. Partition: Two sets or events, A and B, are mutually exclusive (or disjoint) if they have no sample points in common, i.e., if A ∩ B = ∅ . They are collectively exhaustive if together they include all elements, i.e., if AB = S. These concepts can be extended to any number of sets or events. A group of sets that is mutually exclusive and collectively exhaustive is called a partitio.n Some Relationships: • If A ⊆B and B ⊆A, then A = B(anti-symmetric property). • A ⊆ A (reflexive relation). • If A ⊆B and B ⊆C, then A ⊆C (transitivity). • Distributive propertiesof the intersection, union, and complement operators:  A ∩ (B C) = (A ∩B) (A ∩ C) .  A (B ∩ C) = (A B) ∩ (A C) . c c c  (A B) = A ∩ B  (A ∩ B) = A c Bc (calledDe Morgan's Laws, after Augustus De Morgan, 1806- 1871). Prb-5 CEE 3770 Probability These relationships can be extended to any number of sets. PROBABILITY MEASURE– Reading: Sections 2.2 and 2.3 Probability ofan Event: Consider experiment E, sample space S, and event A. The probabilityof event A is a number that measures the likelihood that the event will occur when the experiment is performed. The probability of event A will be denoted by Pr[A]. Axioms of Probability: (a) 0 ≤Pr[A] ≤ 1 for any A. (b) Pr[S] = 1. (c) Suppose events A, B, C, ... are allmutually exclusive, i.e., A ∩B = ∅ , etc. Then Pr[A B C ...] = Pr[A] + Pr[B] + Pr[C] + ... Discussion: Why are 0 and 1 selected as the lower and upper bounds of probability measures? Properties of Probability: The following properties are consequences of the above axioms: (a) For any event A, Pr[A c] = 1 -Pr[A] c where A is the complementof A. (b) If∅ is an impossible event, Pr[∅] = 0. Beware: the converse is not necessarily true –Pr[A] = 0 does not require that A is impossible! (c) For any events A and B, Pr[A B] = Pr[A] + Pr[B] -Pr[A∩ B]. If A and B are mutually exclusive, Pr[A ∩B] = Pr[∅] = 0 and Pr[A B] = Pr[A] + Pr[B]. In general, if 1 , 2 , ...n A are mutually exclusiveand collectively exhaustive(i.e., if ihe As form a partition), then Prb-6 CEE 3770 Probability Pr[A ∩ A] = 0 for all i =/ j, and i j n Pr[A 1 A 2 ... A n (= ∑Pr[A]) i i=1 = Pr[S] = 1 . Recall that c Pr[A] + Pr[A ] = 1 (property (a)), c and note that A and A constitutea partition. Exercise2: Using a Venn diagram, derive an expression for Pr[A B C], as a function of probabilities of events and of intersections of events (Pr[A], Pr[AB], etc.). Exercise 3: A single bolt is selected at random from a box of 10,000. Three different kinds of defects, A, B, and C, are known to occur in these particular bolts. Type A defects occur 1.0 percent of the time, type B defects occur 0.5 percent of the time, and type C 0.75 percent of the time. In addition, it is estimated that 0.25 percent have both A and B defects, 0.30 percent have both A and C, 0.20 percent have both B and C, and 0.10 percent have all three defects. What is the probability that the sample bolt has at least one of the three types of defects? [Hines and Montgomery, 1980, p.32] Definition of Probability: The axioms and properties discussed so far do not tell us how a numeric value can be assigned to an event as its probability (unless A = S or A = ∅ ). However, they do restrict the way in which the assign ment may be made. In practice, proba bility may be assigned on the basis of (1) an analytical consideration of the experi ental conditions, (2) estimates obtained from previous experience, or (3) assumption or subjective belief. See the "Philosophy" section of the supplemental reading list for additional discussion. 1. Definition Using the Concept of Equally Likely Sample Points This definition uses an analytical consideration of expermi ental conditions. Consider: E: Cast a fair die, observe the fac-p number. S: {1, 2, ..., 6} Prb-7 CEE 3770 Probability Let event A be that an even number appears, and event B be that an odd number appears, or, A = {2, 4, 6} and B = {1, 3, 5}. Note that A and B are mutually exclusive (A ∩ B = ∅ ). They are also collectively exhaustive since the union of A and B is the sample space (A B = S). Therefore, Pr[A] + Pr[B] = Pr[S] = 1. Now, because of the experimental condition, it can be as sumed that each of the sample points is equally likely. Since both events A and B contain the same number of equally likely sample points, these two events are also equally likely, i.e., Pr[A] = Pr[B]. Finally, because Pr[A] + Pr[B] = 1, Pr[A] = Pr[B] = 1/2. Consider another example: E: Draw an item from a bin which contains three types of parts, A, B, and C. Observe the type of the item drawn. S: {A, B, C}. Suppose the number of parts in the bin is known by type; the number of parts in the bin is 1,000, of which 300 are type A, 600 type B, and 100 type C. Let A be the event that an item of type A is drawn. Then Pr[A] = (No. of parts of type A)/(Total number of parts) = 300/1000 = 0.30 Exercise4: Show how the above result is obtained using the equal likel hiood concept. Discussion: Quite often, we do not have equally likely outcomes. What if a coin is not true, or a die is not fair? Does the concept apply to the probability that a randomly chosen student will receive an A from a course s/he is taking? Another limitation: This probability definition may be circular. What does "equally likely" mean? Equal probability? See Kyburg, Jr., 1970, Chapter 3 on the Supple mental Reading List for a discussion of this issue. Prb-8 CEE 3770 Probability 2. Definition Based on Relative Frequency This definition is based on past observations and does not require analytical consideration of the likelihood of each outcome. Suppose an experiment is repeated under ideal conditions for n times. Let f be the number of n times event A occurs in the n repetitions. The ratio, f nn, is called the relative frequencyof event A. If n /n approaches some number, say p, as the number of repetitions n increases, then the number p is ascribed to A as its probability, i.e., Pr[A] = lim f/nn= p. n → ∞ In this manner, it is possible to define probabilities for various events, e.g., probability of fatality in a traffic accident, probability that a randomly chosen chip is defetive, etc. Note the difference between Definitions 1 and 2. In Definition 1, the number of possible outcomes in favor of event A is evaluated analytically, while in Definition 2 the number of favor able outcomes is obtained from repeated experiments. Of course Definition 2 cannot be used if empirical observation is not available. 3. Personal (Subjective) Probability The third definition is based on some quantitative measure of personal, subjective belief about the occurrence of an event. This definition is useful for unrepeatable experiments (e.g., a Kentucky Derby race,the 2020 presi- dential election), or for events that have never occurred before (e.g., a person landing on Mars, a nuclear war). Although there are important applica tion areas of this definition (e.g., probabilistic decision analysis), our discussions in this course will be limited to objectively defined probability measures. Probabilistic Models Note also that we can develop models to predict the probability of a certain event, as a function of various explanatory variables. Transportation analysts, for example, often deal with "mode choice models". These models predict the probability that an individual will select a particular travel mode (e.g., auto versus bus) as a function of characteristics of each mode (e.g., travel time and cost) and of the individual (e.g. income, auto ownership). Probabilistic models can combine elements of the 2nd and 3rd approaches to defining probability: the parameters of the model are usually calibrated Prb-9 CEE 3770 Probability based on observations of relative frequency, but in some cases (e.g., forecasting the response to the introduction of a new mode) subjective estimation of one or more parameters may be required. ENUMERATION METHODS– Reading: Section 2.1.4 (CD) Suppose we can identify equally likely sample points of an experiment by inspecting the experimental conditions or by imposing assumptions. Then, as we have seen from Exer cise 4, determining the probability of event A is equivalent to identifying the proportion of equally likely sample points contained in A. Enumerating, or counting, sample points of an experiment thus becomes an important task in evaluating the probability of an event. We will learn five useful enumeration rules. Tree Diagrams: In a simple experiment, a tree diagram may be useful in enumerating sample points and defining the sample space. Exercise 5: Consider Experiment 1a discussed earlier. Develop a tree diagram to enumerate all possible outcomes. Multiplication Principle: Suppose sets A , A 1 ..2, A havk, respectively, n , n 1 .2. ,n ekements. There are n 1 .2. n kays to select a set of elements, one from each set (i.e., an element from A , 1 then an element from A , ..., and finally an element from A). 2 k Exercise 6: Suppose we toss a true coin and cast a true die, and observe the pair of fac e-up outcomes. Determine the number of sample points in the sample space using the mul tiplication rule. Enumerate the sample points by de veloping a tree diagram. Exercise 7: (a) How many different license plates can be generated when they are made up of three letters from the alphabet followed by fournumbers? (b) How many can be generated when no letter or number can be used more than once? PERMUTATIONS AND COMBINATIONS Permutations: A permutation is an arrangement of distinct objects in a dis tinct order. One permu- tation differs from anotherif the content differs,or if the order of arrangement differs. Theorem 1: The number of permutations of N distinct objects is N! . Example: Consider three objects, a, b, and c. There exist 3! = 6 possible arrangements: abc, acb, bac, bca, cab, cba. Prb-10 CEE 3770 Probability Exercise 8: Show that Theorem 1 holds. Theorem 2: Permutations Rule The number of permutations of N distinct objects taken n at a time is N nor PnN = N!/(N -n)! = N× (N-1) × ... × (N-n+1) . Example: Consider four objects, a, b, c, and d. If we consider all permutations taken two at a time, there exist 4!/(4 -2)! = 12 arrangements, which are: ab, ba, ac, ca, ad, da, bc, cb, bd, db, cd, dc. Exercise 9: Show that Theorem 2 holds. Combinations: A combination is an arrangement of distinct objects where one combination dif- fers from anotheronly if the content of the arrangement differs. For example, the six pem r utations shown in the example for Theorem 1 a bove are all from the same combination. Theorem 3: Combinations Rule The number of combinations of N distinct objects taken n at a time is N N nor C N or   = P N / n! =  N!  . n n  n  n!(N − n)!  Exercise 10: Show thatTheorem 3 holds. Example: How many ways are there to select three distinct letters from the alphabet? 26   26!  26×25×24    =   =   = 2,600.  3  3!23!   3×2×1  Theorem 3 can be extended to the following more general statement: Theorem 4: Partitions Rule The number of partitions of N objects into k distinct groups, with iobjects in thei group, is  N   N!  n ,n ,...,n  , or n ! ×n ! ×...×n ! ,  1 2 k   1 2 k  Prb-11 CEE 3770 Probability where N = n + 1 + ..2 + n . In tkhis theorem, the k types are distinguished when enumerating arrangements, but objects within each type are not distin guished (i.e., they are thought of as combinations). It is often useful to think of the Partitions Rule as answer ing the question, "How many sequences containing n 11s', n 22s', ..., and k 'ks' are there?" Exercise 11: A group of 15 engineers is to be divided into three teams of five engineers each, to work on three distinct projects. (a) In how many different ways can this assignment be made? (b) Suppose there are 5 engineers who are qualified as project leaders and that each project team must have one project leader. In how many ways can the 15 engineers be divided into 3 teams of 5 engineers, each having one designate d project leader? Exercise 12: Show that Theorem 3 (the Combinations Rule) is the special case of Theorem 4 (the Partitions Rule) for k=2. Why? "MultipleCombinations" Rule: Suppose a set of N elements is partitioned into k subsets, with n iobjects in the ith subset. The number of ways of selecting j eliments from each subset i (i.e., j ele1ents from the n eleme1ts in set 1, 2 elements from the n i2 set2, ... , jkelements from n ) ks: n1 n 2 n k j   j  …  j  .  1  2  k Exercise 13: Suppose the Homecoming Court at a small college is to consist of 1 member from each of the senior, junior, sophomore, and freshman classes. Assuming that men and women are equally eligible, and that there are 125 sen iors, 130 juniors, 120 sophomores, and 150 freshmen, (a) how many possibilities are there for the Homecoming Court composition? (b) How many possibilities are there if the court is to con tain three seniors, three juniors, two sophomores, and twofreshmen? Exercise 14: How many different license plates are there containing three letters and four numbers in any order,if (a) all letters and numbers are distinct; and (b) they are not necessarily distinct? Prb-12 CEE 3770 Probability Exercises: Combinatorial Problems The following exercise problems offer examples in which the equally likely sample points are enumerated(counted, not listed!)and used to evaluate the probability of an event. Exercise15: A production lot contains 100 toggle switches, of which three are known to be defective. A sample of five switches is drawn randomly from the lot, without replacement. Dete- r mine the probabilities that (a) the sample contains no defective switch, (b) the sample contains exactly two defective switches, and (c) the sample contains at least one defective switch. Exercise16: Consider poker hands of 5 cards being dealt from an ordinary well-shuffled deck of 52 cards. Determine the probability (a) that a flush (all cards of one suit) is dealt, (b) that the queen of spades is one of 5 cards dealt, (c) that exactly two queens are dealt, and (d) that exactly one pair is dealt. Exercises: Additional Problems Exercise 17: Buffon's Needle. A hardwood floor is made of boards 4 inches wide. A needle of length 2 inches is tossed in the air and lands on the floor. What is the probability the needle comes to rest crossing a joint between the floorboards? (Answer: 1 π/) Exercise 18: Eight tires of different brands are ranked from one to eight (best to worst) according to mileage p erformance. If four of these tires are chosen at random by a customer, find the probability that the best tire among those selected by the customer is actually ranked third among the original eight. [Mendenhall & Scheaffer, 1973, p.66](Answer: 1/7) Exercise 19: Nine people are going on a skiing trip in 3 cars that will hold 2, 4, and 5 persons, respectively. In how many ways is it possible to transport the 9 people to the ski lodge using all cars? [Walpole & Myers, 1985, p.18](Answer: 4410) Exercise 20: A machine for producing a new experimental electronic tube generates defective tubes from time to time in a random manner. The supervising engineer for a particular machine has noticed that defective items seem to be grouping (hence appearing in a non random manner) and thereby suggesting a malfunction in some part of the machine. One test for nonrandomness is based on the number of "runs" of defective and nondefective items (a run is an unbroken sequence of either defective or nondefective items). The smaller the number of runs, the greater will be the Prb-13 CEE 3770 Probability amount of evidence indicating nonrandomness. Of 12 tubes drawn from the machine, the first 10 were nondefective and the last 2 were defective, (NNNNNNNNNNDD). Assuming randomness, (a) What isthe probability of observing the arrangement shown above (resulting in two runs) given that 10 of the 12 tubes are nodefective? (Answer: 1/66) (b) What is the probability of observing two runs?(Answer: 2/66) (c) What is the probability that thenumber of runs, R, is no more than 3?(Answer: 12/66) [Mendenhall & Scheaffer, 1973, p.64] CONDITIONAL PROBABILITY– Reading: Sections 2.4, 2.5.1 Quite often we are interested in the probability of an event given the knowledge that another event has taken place or has not taken place. For example, the probability that an automobile will be involved in an accident, given that the pavement is wet. Definition: The probability that event A occurs given that event B has occurred, is called the conditional probabilityof event A given event B, and is denoted by Pr[A|B]. This probability is defined as: Pr[A| B] = Pr[A ∩B]/Pr[B] if Pr[B] > 0 (Pr[A| B] is undefined if Pr[B] = 0). Rewriting this formula for conditional probability, Pr[A∩ B] = Pr[A|B] Pr[B] = Pr[B|A] Pr[A] . In words, the probability that A and B both occur is the product of the probability that A occurs and the conditional probability that B occurs given A has occurred. From this, Pr[B| A] / Pr[A |B] = Pr[B] / Pr[A]. Prb-14 CEE 3770 Probability Exercise 21: In an experiment to study the dependence of hypertension on smoking habits, the following data were collected on 180 individuals: ========================================================== Moderate Heavy Non-Smokers Smokers Smokers ========================================================== Hypertension 21 36 30 No Hypertension 48 26 19 ========================================================== If one of these individuals is selected at random, find the probability that the person is: (a) experiencing hypertension, given that the person is a heavy smoker; (b) a non-smoker, given that the person is experiencing no hypertension. (c) Does Pr[NS |NHT] = Pr[NHT |NS]? Find (d) Pr[NHT |HS], (e) Pr[NS|HT]. c An important relationship is: Pr[A|B] = 1 -Pr[A|B] (see Exercise 27). Generalization, Chain Rule(not in book, but important!) What if we know two things – i.e., that B and C have oc curred? Replacing B in the relation Pr[A|B] = Pr[A ∩B] / Pr[B] by B∩ C, we obtain Pr[A|B ∩ C] = Pr[A ∩B ∩ C] / Pr[B ∩C] (we shall hereafter use the expression, Pr|B,C], instead of Pr[A|B ∩ C]). Just like we saw that Pr[A|B] Pr[B] is a convenient formula for Pr[A ∩ B], we want a convenient formula for Pr[A ∩B ∩ C]. From above we have Pr[A∩ B ∩ C] = Pr[A|B, C] Pr[B ∩C] = Pr[A|B, C] Pr[B|C] Pr[C]. In general, Pr[A n|A1,A2,...,n-1 = Pr[A 1 A ∩2... ∩ A ] n Pr[A ∩1A ∩ 2.. ∩ A ] n-1 and Pr[A 1 A ∩ 2.. ∩ A ] n Prb-15 CEE 3770 Probability = Pr[A ∩1A ∩ .2. ∩ A ] Prn-1|A ,A n..1,A 2 n-1 Using this relation recursively, we obtain Pr[A ∩ A ∩ ... ∩ A ] 1 2 n = Pr[A ]1r[A |A2] 1r[A |A3,A 1 .2. Pr[A |A nA 1..2,A ] n-1 This relation, often called the"chain rule",is useful in many applications. Exercise 22: Suppose that a recent census found that 51% of Atlanta adults are male; 55% of Atlanta adult males are Democrats; and 30% of adult male Democrats in Atlanta are age 45 or older. In general, 40% of all adult Democrats in Atlanta are male. An Atlanta adult is selected at random. (a) What is the sample space for this experiment? (b) Define all relevant events and express all relevant numbers as probab itiiles of events. What is the probability that the selected person is (c) a male Democrat age 45 or older? (d) a Democrat? (e) a Democrat, given that it is a female? STATISTICAL INDEPENDENCE– Reading: Section 2.6 When the occurrence of event A does not influence the occurrence of event B, these two events are said to be statistically independent. Formally: Two events A and B are statistically independent if Pr[B|A] = Pr[B] or Pr[A|B] = Pr[A]. (You can easily prove that if one equations i true the other is automatically true). Multiplicative Rule for Independent Events: Events A and B are independent if and only if Pr[A∩ B] = Pr[A] Pr[B]. The following is an excerpt from Breipohl (1970) which offers us useful insights about statist ical independence: Prb-16 CEE 3770 Probability “... sometimes events are not statistically independent even though no direct cause -and-effect relation may be seen. For example, consider the events that represent the failure of two resistors. It is easily seen that if the two resis tors are used in parallel, then the failure of one will change the current in the other, so one would expect these two events to be statis tically dependent (not independent). However, suppose they are not used in the same circuit but are used in the sa me piece of equipment. Then the failure of any one may change the ambient temperature, thereby changing the probability of failure of the other .... The question of statistical independence is often a difficult one; the best one can do is to try toason and collect data. It should be remembered that the defining equations are the criteria, not some vague words about cause and effect.” Exercise 23: Consider the data of Exercise 21. Are the events hypertension and heavy smoker independent? Exercise 24: In a new braking device designed to prevent automobile skids, there is a conside arble amount of electronic and hydraulic hardware. The entire system may be broken down into three series of subsystems that operate independently: (E) an electro nics system, (H) a hydraulic system, and (M) a mechanical activator. On a particular braking system the relia bility of these units are approximately, 0.995, 0.993, and 0.994, respec tively. Estimate the system reliability. [Hines and Montgomery, 1980, p.34] Note: Such a system is called a linear or series system . If any component breaks down, the entire system fails. So all components must work for the system to function. Forparallel components, by contrast, if any one of them works, the system functions. Exercise 25: Consider the series -parallel assembly shown below. The values R i are the reliabilities for the five components shown, that is, Ri= probability that unit i will function proper- ly. The components operate (and fail) ina mutually independent manner and the assembly fails only when the path from A to B is broken. Express the assembly reliability as a function of R 1 through R 5. [Hines and Montgomery, 1980, p.60] Prb-17 CEE 3770 Probability Generalization: The concept of independence can be extended to n events. The events A, A ,..., 1 2 A nre statistically independent if and only if the intersections of all possible groups of events obey the multiplicative rule. That is, iff Pr[A i(1) A i(2) ... ∩ A i(k) = Pr[A ]i(1)A ] .i(2)r[A ] i(k) for any k, k = 2,..., n, and for any i(1) < i(2) < ... < i(k), 1i(1), i(k) ≤n. If n = 3, the condition for independence becomes Pr[A ∩1A ] =2Pr[A ] Pr1A ] an2 Pr[A 1∩ A ]3= Pr[A ] 1r[A ] 3nd Pr[A ∩2A ] =3Pr[A ] Pr2A ] an3 Pr[A 1∩ A ∩2A ] =3Pr[A ] Pr1A ] Pr[2 ] . 3 Exercise 26: If n=5, how many different conditions must be checked before independence is verified? Example: Suppose two honest coins are tossed and the up- faces recorded. Let A be the event "the first coin is a head," let B be the event "the second coin is a head," and let C be the event "the coins match" (both heads or both tails). A and B are independent events, and also A and C, as well as B and C, are independent events. That is, the events A, B, and C are pairwise independent. To see this, we can enumerate equally likely ouc tomes of the experiment, and obtain Pr[A] = Pr[A|B] = Pr[A|C] = 1/2 Pr[B] = Pr[B|A] = Pr[B|C] = 1/2 Pr[C] = Pr[C|A] = Pr[C|B] = 1/2. But observe that Pr[A∩ B ∩ C] = Pr[{HH}] = 1/4, while 3 Pr[A] Pr[B] Pr[C] = (1/2) = 1/8. Thus the events A, B, and C are not independent, although they are pairwise independent (Clarke and Disney, 1985, p.26). In this example (where n = 3), h t e first three of the four conditions shown above are met, but not the last one. Prb-18 CEE 3770 Probability A Note on Independence and Exclusiveness Statistical independence and exclusiveness are entirely different concepts. It is not true that two mutually exclusive (or "djoint") events are independent. To the contrary, MUTUALLY EXCLUSIVE EVENTS ARE STATISTICALLY DEPENDENT , AND (EQUIVALENTLY) INDEPENDENT EVENTS MUST INTERSECT! Mathematically, the first statementis true since, if A ∩B = ∅ but A≠ ∅ and B≠ ∅ , then Pr[A∩ B] = 0 Pr[A] Pr[B]. Intuitively, it is true because if two events are M.E., then knowing that B has occurred tells you A cannot occur. This violates the concept of independence, which holds that knowinghas occurred does not affect the probability of A occurring. Thus, for M.E. events, B|] = 0 ≠Pr[A]. Conversely, just because two events intersect doesn't mean they are dependent. See examples above (in which A ∩B ≠ ∅, but A and B are independent ) and below. Unlike mutual exclusivity, independence "cannot be shown on or gleaned from a Venn diagram, and you cannot trust your intuition" (M & S, p. 110). The only way to be sure of independence is to calculate the appropriate probabilities. Example: For example, consider an urn with 100 marbles. Fifty of the marbles are blue and the other 50 are red. One half (25) of the blue marbles are even-numbered and the other half odd- numbered; similarly one half of red marbles are even-numbered and the other half odd-numbered. So there are 50 even-numbered marbles and 50 odd-numbered marbles. Consider an experiment where a marble is drawn at random from the urn. Let events B and R be B = {marble is blue} and R = {marble is red}. Then B ∩R = ∅ , i.e., B and R are mutually exclusive. But Pr[B∩ R] = Pr[∅] = 0 ≠ Pr[B] Pr[R] = 0.25. Therefore B and R are not independent. Now consider events E and O defined as E = {marble is even-numbered} and O = {marble is odd numbered} . Prb-19 CEE 3770 Probability Then, Pr[B∩ E] = 25/100 = Pr[B] Pr[E], i.e., B and E are independent, butthey are not exclusive. TOTAL PROBABILITY– Reading: Sections 2.5.2, 2.7 In calculating the probability of an event E, it is often convenient to partition E into several subsets, and to express Pr[E] as the sum of the probabilities of those sub-events. The specific partition we choose will be based on some other events that we expect will affect Pr[E]. To make sure that we calculate thetotal probabilityof E, we specify that those otherevents must form a partition of S. Formally, consider events A, A1, .2., A ank suppose they are apartition, i.e., Pr[A] > 0 ior all i, A i∩ A =j∅ for all i, j;= j, and A 1 A 2 ... A k S. These events are mutually exclusiv e and collectively exhaustive. Using the multiplicative rule, Pr[E∩ A i] = Pr[E|A] ir[A] i for any event E. Now noting that E = E ∩S = E ∩[A 1 A 2 ... Ak] = [E ∩A ] 1 [E ∩ A 2 ... [E ∩ A ]k, we have Pr[E] = Pr[E ∩A ] +1Pr[E ∩A ] + ..2 + Pr[E ∩A ] k = Pr[E|A ]1Pr[A ]1+ Pr[E|A ] 2r[A ] +2... + Pr[E|A ] Pk[A ], k or, Pr[E] = Σ Pr[E|A i] Pr[A]i i Prb-20 CEE 3770 Probability This is called the total probabilityformula. The equation holds for any event E as long as the As form a partition. Example: Consider the data shown in Exercise 21. Events {HT, NHT} form a partition, and so do events {NS, MS, HS}. From the total probability formula, we obtain, e.g., Pr[NS] = Pr[NS|HT] Pr[HT] + Pr[NS|NHT] Pr[NHT] = (21/87) (87/180) + (48/93) (93/180) = (21 + 48) / 180 = 69/180. Similarly, Pr[HT] = Pr[HT|NS]Pr[NS] + Pr[HT|MS]Pr[MS] + Pr[HT|HS]Pr[HS] = (21/69)(69/180) + (36/62)(62/180) + (30/49)(49/180) = (21 + 36 + 30) / 180 = 87/180. Exercise 27: Use the total probability idea to show that, if Pr[B] > 0, c Pr[A|B] = 1 -Pr[A |B] . Bayes' Theorem Let Pr[E] ≠ 0 and A i i = 1, ..., k be apartition with Pr[i] > 0 for all i. Then, Pr[A |E] = Pr[E ∩A] / Pr[E] = Pr[E|A] Pr[A] / Pr[E]. i i i i But from the total probability theorem, Pr[E] = ΣPr[E|A] Pr[A]. i i i Therefore, Pr[E| Ai] Pr[ A i] Pr[ A i|E]= . ∑ Pr[E| A j Pr[ A j j This is called Bayes' formula or Bayes' rule, and dates back to 1763. Observe that Bayes' rule reverses the conditionality. Both the total probability formula and Bayes' rule are useful in many applications, as the following exercise problems indi cate. Prb-21 CEE 3770 Probability Exercise 28: There are three plants, B , B , and B , which produce the same product with 1 2 3 different defect rates (see table below). The total production is 1,000 units per day. At the end of the day, the products from the respective plants are mixed and stored. Suppose one item is randomly selected from the pool of 1,000 products. Determine: (a) the probability that the item is from P1ant B, (b) the probability that the item is defective by applying the total probabili,ndformulaa (c) the probability that the item is from P1ant B, given it is defective. PLANT PRODUCTION DEFECT RATE =================================== B1 300 10% B2 500 8% B 200 20% 3 =================================== Total 1000 (Units/day) Exercise 29: A binary communication channel is a system which carries data in the form of one of two types of signals, e.g., either "0" or "1". Because of noise, a "0" transmitted is sometimes received as a "1" and a "1" transmitted is sometimes received as a "0". It is known for a certain binary communication channel that the probability that a transmitted "0" is received as a "0" is 0.95, and the probability that a transmittedceived as a "1" is 0.90. It is also known that the probability that a "0" is transmitted is 0.40. Find: (a) the probability that a "1" is received, and (b) the probability that a "1" was transmitted given a "1" was received. [Breipohl, 1970, p.35] Prb-22 CEE 3770 Random Variables RANDOM VARIABLES– Reading: Section 2.8 At this point, we have a complete probability structure: we have discussed a random experiment, sample points (or outcomes of the experiment), a set of outcomes that define the sample space, certain subsets of the sample space defining events, and an assignment of a number to each event which satisfies the axioms of probability (we have used the con cepts of equal likelihood and relative frequency). Thus far we have constructed the sample space as aes t of outcomes, which are not always quantitative (e.g., heads and tails) leading to non-numerical sample spaces. For mathe- matical purposes, this is inconvenient. This leads to the disussion ofrandom variables. Definition: A random variable is a function (or a "rule") that assigns a real number to each point in the sample space(so the argumentof the function is a sample poin) t. We will consider only those cases where themapping from a sample point to a real value is uniqu.e We do not require uniqueness in the other direction, i.e., the mapping from the real axis to the sample space need not be unique (that is, a given real value can be associated with 2 or more sample points, but any given sample point maps only to one real valu)e . In the following discussio


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