1. Chapter 3 Basic Concepts in Statistics and Probability

2. 3.1 Probability Definition: An experiment is a process that results in an outcome that cannot be predicted in advance with certainty. Examples:  rolling a die  tossing a coin  weighing the contents of a box of cereal. 2

3. Sample Space Definition: The set of all possible outcomes of an experiment is called the sample space for the experiment. Examples: For rolling a fair die, the sample space is {1, 2, 3, 4, 5, 6}. For a coin toss, the sample space is {heads, tails}. Imagine a hole punch with a diameter of 10 mm punches holes in sheet metal. Because of variation in the angle of the punch and slight movements in the sheet metal, the diameters of the holes vary between 10.0 and 10.2 mm. For this experiment of punching holes, a reasonable sample space is the interval (10.0, 10.2). 3

4. More Terminology Definition: A subset of a sample space is called an event. A given event is said to have occurred if the outcome of the experiment is one of the outcomes in the event. For example, if a die comes up 2, the events {2, 4, 6} and {1, 2, 3} have both occurred, along with every other event that contains the outcome “2”. 4

5. Combining Events The union of two events A and B, denoted A  B, is the set of outcomes that belong either to A, to B, or to both. In words, A  B means “A or B.” So the event “A or B” occurs whenever either A or B (or both) occurs. Example: Let A = {1, 2, 3} and B = {2, 3, 4}. What is A  B? 5

6. Intersections The intersection of two events A and B, denoted by A  B, is the set of outcomes that belong to A and to B. In words, A  B means “A and B.” Thus the event “A and B” occurs whenever both A and B occur. Example: Let A = {1, 2, 3} and B = {2, 3, 4}. What is A  B? 6

7. Complements The complement of an event A, denoted A c, is the set of outcomes that do not belong to A. In words, A c means “not A.” Thus the event “not A” occurs whenever A does not occur. Example: Consider rolling a fair sided die. Let A be the event: “rolling a six” = {6}. What is A c = “not rolling a six”? 7

8. Mutually Exclusive Events Definition: The events A and B are said to be mutually exclusive if they have no outcomes in common. More generally, a collection of events A 1, A 2, …, A n is said to be mutually exclusive if no two of them have any outcomes in common. Sometimes mutually exclusive events are referred to as disjoint events. 8

9. Probabilities Definition:Each event in the sample space has a probability of occurring. Intuitively, the probability is a quantitative measure of how likely the event is to occur. Given any experiment and any event A: The expression P(A) denotes the probability that the event A occurs. P(A) is the proportion of times that the event A would occur in the long run, if the experiment were to be repeated over and over again. 9

10. Axioms of Probability 1.Let S be a sample space. Then P(S) = 1. 2.For any event A,. 3.If A and B are mutually exclusive events, then. More generally, if are mutually exclusive events, then 10

11. A Few Useful Things For any event A, P(A C ) = 1 – P(A). Let  denote the empty set. Then P(  ) = 0. If S is a sample space containing N equally likely outcomes, and if A is an event containing k outcomes, then P(A) = k/N. Addition Rule (for when A and B are not mutually exclusive): 11

12. Conditional Probability and Independence Definition: A probability that is based on part of the sample space is called a conditional probability. Let A and B be events with P(B)  0. The conditional probability of A given B is 12

13. Conditional Probability 13 Venn Diagram

14. Independence Definition: Two events A and B are independent if the probability of each event remains the same whether or not the other occurs. If P(A)  0 and P(B)  0, then A and B are independent if P(B|A) = P(B) or, equivalently, P(A|B) = P(A). If either P(A) = 0 or P(B) = 0, then A and B are independent. These concepts can be extended to more than two events. 14

15. The Multiplication Rule If A and B are two events and P(B)  0, then P(A  B) = P(B)P(A|B). If A and B are two events and P(A)  0, then P(A  B) = P(A)P(B|A). If P(A)  0, and P(B)  0, then both of the above hold. If A and B are two independent events, then P(A  B) = P(A)P(B). 15

16. Extended Multiplication Rule If A 1, A 2,…, A n are independent results, then for each collection of A j1,…, A jm of events In particular, 16

17. Example A system contains two components, A and B, connected in series. The system will function only if both components function. The probability that A functions is 0.98 and the probability that B functions is 0.95. Assume A and B function independently. Find the probability that the system functions. 17

18. Example A system contains two components, C and D, connected in parallel. The system will function if either C or D functions. The probability that C functions is 0.90 and the probability that D functions is 0.85. Assume C and D function independently. Find the probability that the system functions. 18

19. Example P(A)= 0.995; P(B)= 0.99 P(C)= P(D)= P(E)= 0.95 P(F)= 0.90; P(G)= 0.90, P(H)= 0.98 19

20. Random Variables Definition:A random variable assigns a numerical value to each outcome in a sample space. Definition:A random variable is discrete if its possible values form a discrete set. 20

21. Probability Mass Function The description of the possible values of X and the probabilities of each has a name: the probability mass function. Definition:The probability mass function (pmf) of a discrete random variable X is the function p(x) = P(X = x). The probability mass function is sometimes called the probability distribution. 21

22. Probability Mass Function Example 22

23. Cumulative Distribution Function The probability mass function specifies the probability that a random variable is equal to a given value. A function called the cumulative distribution function (cdf) specifies the probability that a random variable is less than or equal to a given value. The cumulative distribution function of the random variable X is the function F(x) = P(X ≤ x). 23

24. More on a Discrete Random Variable Let X be a discrete random variable. Then The probability mass function of X is the function p(x) = P(X = x). The cumulative distribution function of X is the function F(x) = P(X ≤ x).., where the sum is over all the possible values of X. 24

25. Mean and Variance for Discrete Random Variables The mean (or expected value) of X is given by, where the sum is over all possible values of X. The variance of X is given by The standard deviation is the square root of the variance. 25

26. Example 26 Probability mass function will balance if supported at the population mean

27. The Probability Histogram When the possible values of a discrete random variable are evenly spaced, the probability mass function can be represented by a histogram, with rectangles centered at the possible values of the random variable. The area of the rectangle centered at a value x is equal to P(X = x). Such a histogram is called a probability histogram, because the areas represent probabilities. 27

28. Probability Histogram for the Number of Flaws in a Wire The pmf is: P(X = 0) = 0.48, P(X = 1) = 0.39, P(X=2) = 0.12, and P(X=3) = 0.01. 28

29. Probability Mass Function Example 29

30. Continuous Random Variables A random variable is continuous if its probabilities are given by areas under a curve. The curve is called a probability density function (pdf) for the random variable. Sometimes the pdf is called the probability distribution. The function f(x) is the probability density function of X. Let X be a continuous random variable with probability density function f(x). Then 30

31. Continuous Random Variables: Example 31

32. Computing Probabilities Let X be a continuous random variable with probability density function f(x). Let a and b be any two numbers, with a < b. Then In addition, 32

33. More on Continuous Random Variables Let X be a continuous random variable with probability density function f(x). The cumulative distribution function of X is the function The mean of X is given by The variance of X is given by 33

34. Two Independent Random Variables 34

35. Variance Properties If X 1, …, X n are independent random variables, then the variance of the sum X 1 + …+ X n is given by If X 1, …, X n are independent random variables and c 1, …, c n are constants, then the variance of the linear combination c 1 X 1 + …+ c n X n is given by 35

36. More Variance Properties If X and Y are independent random variables with variances, then the variance of the sum X + Y is The variance of the difference X – Y is 36

37. Independence and Simple Random Samples Definition: If X 1, …, X n is a simple random sample, then X 1, …, X n may be treated as independent random variables, all from the same population. 37

38. Properties of If X 1, …, X n is a simple random sample from a population with mean  and variance  2, then the sample mean is a random variable with The standard deviation of is 38

39. 3.2 Sample versus Population Definitions:  A population is the entire collection of objects or outcomes about which information is sought.  A sample is a subset of a population, containing the objects or outcomes that are actually observed.  A simple random sample (SRS) of size n is a sample chosen by a method in which each collection of n population items is equally likely to comprise the sample, just as in the lottery.

40. Sampling (cont.) Definition: A sample of convenience is a sample that is not drawn by a well-defined random method. Things to consider with convenience samples:  Differ systematically in some way from the population.  Only use when it is not feasible to draw a random sample. 40

41. Simple Random Sampling A SRS is not guaranteed to reflect the population perfectly. SRS’s always differ in some ways from each other; occasionally a sample is substantially different from the population. Two different samples from the same population will vary from each other as well.  This phenomenon is known as sampling variation. 41

42. Tangible Population The populations that consist of actual physical objects – customers, blocks, balls are called tangible populations. Tangible populations are always finite. After we sample an item, the population size decreases by 1. 42

43. More on Simple Random Sampling 43 Definition: A conceptual population consists of items that are not actual objects. For example, a geologist weighs a rock several times on a sensitive scale. Each time, the scale gives a slightly different reading. Here the population is conceptual. It consists of all the readings that the scale could in principle produce.

44. Simple Random Sampling (cont.) 44 The items in a sample are independent if knowing the values of some of the items does not help to predict the values of the others. Items in a simple random sample may be treated as independent in most cases encountered in practice. The exception occurs when the population is finite and the sample comprises a substantial fraction (more than 5%) of the population.

45. Types of Sampling Weighted Sampling Stratified Random Sampling Cluster Sampling 45

46. 3.3 Location Measures used to describe “location” of data: (Measure of center) or (Measure of central tendency) Median Mean (Average) Robust estimators: Trimmed average: 10% of the observations in a sample are trimmed from each end

47. 3.4 Variation Variation: Natural cause Assignable causes Measures of variation: Range (using only the extreme values) Variance Standard deviation Covariance

48. Variation Calculation (3.1) (3.2) (3.3)

49. 3.5 Discrete Distributions Random Variable: “Something that varies in a random manner” Discrete Random Variable: “Random variable that can assume only a finite number of possible values (usually integers)

50. Discrete Random Variable Example Experiment: Tossing a single coin twice and recording the number of heads observed Repeated 16 times X= number of heads observed in each experiment 0 2 1 1 2 0 0 1 2 1 1 0 1 1 0 1 1 2 0 Empirical distribution Theoretical distribution

51. 3.5.1 Binomial Distribution We use the Bernoulli distribution when we have an experiment which can result in one of two outcomes. One outcome is labeled “success,” and the other outcome is labeled “failure.” The probability of a success is denoted by p. The probability of a failure is then 1 – p. Such a trial is called a Bernoulli trial with success probability p. 51

52. Examples of Bernoulli Trials 1.The simplest Bernoulli trial is the toss of a coin. The two outcomes are heads and tails. If we define heads to be the success outcome, then p is the probability that the coin comes up heads. For a fair coin, p = 1/2. 2.Another Bernoulli trial is a selection of a component from a population of components, some of which are defective. If we define “success” to be a defective component, then p is the proportion of defective components in the population. 52

53. Binomial Distribution If a total of n Bernoulli trials are conducted, and  The trials are independent.  Each trial has the same success probability p.  X is the number of successes in the n trials. then X has the binomial distribution with parameters n and p, denoted X ~ Bin(n,p). 53

54. Probability Mass Function of a Binomial Random Variable If X ~ Bin(n, p), the Probability Mass Function of X is 54 (3.5)

55. Binomial Probability Histogram 55 (a) Bin(10, 0.4) (b) Bin(20, 0.1)

56. Example The probability that a newborn baby is a girl is approximately 0.49. Find the probability that of the next five single births in a certain hospital, no more than two are girls. 56

57. Another Use of the Binomial Assume that a finite population contains items of two types, successes and failures, and that a simple random sample is drawn from the population. Then if the sample size is no more than 5% of the population, the binomial distribution may be used to model the number of successes. 57

58. Example A lot contains several thousand components, 10% of which are defective. Nine components are sampled from the lot. Let X represent the number of defective components in the sample. Find the probability that exactly two are defective. 58

59. Software Functions for Binomial Probabilities Excel: BINOM.DIST(number_s, trials, probability_s, cumulative) Minitab: Calc  Probability Distributions  Binomial 59

60. Example Of all the new vehicles of a certain model that are sold, 20% require repairs to be done under warranty during the first year of service. A particular dealership sells 14 such vehicles. What is the probability that fewer than five of them require warranty repairs? 60

61. Mean and Variance of a Binomial Random Variable  E(X) = np  E(Bernoulli Trial)=(1)p+(0)(1-p)=p  Var(X) = np(1 – p)  Var(Bernoulli Trial)=(1-p) 2 p+(0-p) 2 (1-p) =(1-2p+p 2 )p+p 2 (1-p) =p-2p 2 +p 3 +p 2 -p 3 =p-p 2 =p(1-p) 61

62. 3.5.2 Beta-Binomial Distribution 62

63. 3.5.3 Poisson Distribution  One way to think of the Poisson distribution is as an approximation to the binomial distribution when n is large and p is small.  It is the case when n is large and p is small that the mass function depends almost entirely on the mean np, and very little on the specific values of n and p.  We can therefore approximate the binomial mass function with a quantity λ = np; this λ is the parameter in the Poisson distribution. 63

64. Probability Mass Function, Mean, and Variance of Poisson Dist.  If X ~ Poisson(λ), the probability mass function of X is  Mean:  X = λ  Variance: Note: X must be a discrete random variable and λ must be a positive constant. 64 (3.6)

65. Poisson Probability Histogram 65 Figure 4.2 (a) Poisson(1) (b) Poisson(10)

66. Poisson Probabilities Excel: POISSON.DIST(x, mean, cumulative) Minitab: Calc  Probability Distributions  Poisson 66

67. Example Particles are suspended in a liquid medium at a concentration of 6 particles per mL. A large volume of the suspension is thoroughly agitated, and then 3 mL are withdrawn. What is the probability that exactly 15 particles are withdrawn? 67

68. 3.5.4 Geometric Distribution Geometric distribution and the negative binomial distribution are referred as “waiting time” distributions. It deals with the number of trials required for a single success. Outcomes are either success/failure. Trial continues until success (defect) occurs for the first time. –Useful for manufacturing where the line will be shut down for recalibration upon first defect.

69. Geometric Distribution Geometric Distribution: n: The number of trials required to produce 1 success in a geometric experiment. p: The probability of success on an individual trial. 1- p: The probability of failure on an individual trial.

70. Geometric Distribution Mean and Variance  : the average no. of trials required to produce 1 success

71. Geometric Distribution Example Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. What is the probability that Bob makes his first free throw on his fifth shot? Solution: Probability of success (p) is 0.70, the number of trials (x) is 5, and the number of successes (r) is 1. We enter these values into the geometric formula.

72. Geometric Distribution Example Military contractor is producing nuts that must be within.04 mm of specified diameter. If nut exceeds the limit the line must be shut down and adjusted. The probability that the diameter of a nut will exceeds the allowable error is.0014. What is the probability the machine will be shut down exactly after the 100 th nut is produced? What is the probability the machine will be shut down exactly after the 200 th nut is produced?

73. 3.5.5 Negative Binomial Distribution A negative binomial experiment is a statistical experiment that has the following properties:statistical experiment The experiment consists of x repeated trials. Each trial can result in just two possible outcomes, a success and a failure. The probability of success, denoted by P, is the same on every trial. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.independent The experiment continues until r successes are observed, where r is specified in advance.

74. Negative Binomial Distribution A negative binomial random variable is the number X of repeated trials to produce r successes in a negative binomial experiment. The negative binomial distribution is also known as the Pascal distribution.

75. Negative Binomial Distribution n: The number of trials required to produce r successes in a negative binomial experiment. r: The number of successes in the negative binomial experiment. p: The probability of success on an individual trial. 1-p: The probability of failure on an individual trial.

76. Negative Binomial Distribution Mean and Variance  : the average no. of trials required to produce r successes

77. Negative Binomial Distributions Example Bob is a high school basketball player. He is a 70% free throw shooter. That means his probability of making a free throw is 0.70. During the season, what is the probability that Bob makes his third free throw on his fifth shot? Solution: The probability of success (p) is 0.70, the number of trials (x) is 5, and the number of successes (r) is 3.

78. 3.5.6 Hypergeometric Distribution A sample of size n is randomly selected without replacement from a population of N items. In the population, r items can be classified as successes, and N - r items can be classified as failures. A hypergeometric random variable, x, is the number of successes that result from a hypergeometric experiment

79. Hypergeometric Probability Distribution Where N = total number of elements in the population D = number of success in the population N-D= number of failures in the population n = number of trials (sample size) x= number of successes in trial n-x= number of failures in n trials

80. Hypergeometric Distribution Mean and Variance Where p= r/N

81. Hypergeometric Probability Distribution Example Suppose we select 5 cards from an ordinary deck of playing cards. What is the probability of obtaining 2 or fewer hearts? Solution: N = 52; since there are 52 cards in a deck. r = 13; since there are 13 hearts in a deck. n = 5; since we randomly select 5 cards from the deck. x = 0 to 2; since our selection includes 0, 1, or 2 hearts. We plug these values into the hypergeometric formula as follows:

82. Hypergeometric Probability in MINITAB Acceptance testing of ice cream cones Ice cream parlor checks a batch of 400 waffle cones by checking 50 of them. They will not buy them if more than 3 cones are broken. What is the probability that the parlor will buy the cones if 35 of the 400 cones are broken. –Define N, n, D, N-D, x –In MINITAB select: Calc-> Probability Distributions - > Hypergeometric