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5 - 1 © 1998 Prentice-Hall, Inc. Chapter 5 Continuous Random Variables.

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1 5 - 1 © 1998 Prentice-Hall, Inc. Chapter 5 Continuous Random Variables

2 5 - 2 © 1998 Prentice-Hall, Inc. Learning Objectives 1.Define continuous random variable 2.Describe the normal random variables 3.Calculate probabilities for Normal random variables 4.Approximate the binomial distribution using the normal distribution

3 5 - 3 © 1998 Prentice-Hall, Inc. Data Types

4 5 - 4 © 1998 Prentice-Hall, Inc. Continuous Random Variables

5 5 - 5 © 1998 Prentice-Hall, Inc. Continuous Random Variables 1. Random variable A numerical outcome of an experiment A numerical outcome of an experiment Weight of a student (e.g., 115, 156.8, etc.) Weight of a student (e.g., 115, 156.8, etc.) 2. Continuous random variable Whole or fractional number Whole or fractional number Obtained by measuring Obtained by measuring Infinite number of values in interval Infinite number of values in interval Too many to list like discrete variable Too many to list like discrete variable

6 5 - 6 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples

7 5 - 7 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values

8 5 - 8 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people

9 5 - 9 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people Weight 45.1, 78,...

10 5 - 10 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people Weight 45.1, 78,... Measure part life

11 5 - 11 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people Weight 45.1, 78,... Measure part life Hours 900, 875.9,...

12 5 - 12 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people Weight 45.1, 78,... Measure part life Hours 900, 875.9,... Ask food spending

13 5 - 13 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people Weight 45.1, 78,... Measure part life Hours 900, 875.9,... Ask food spending Spending 54.12, 42,...

14 5 - 14 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people Weight 45.1, 78,... Measure part life Hours 900, 875.9,... Ask food spending Spending 54.12, 42,... Measure time between arrivals

15 5 - 15 © 1998 Prentice-Hall, Inc. Continuous Random Variable Examples ExperimentRandom Variable Possible Values Weigh 100 people Weight 45.1, 78,... Measure part life Hours 900, 875.9,... Ask food spending Spending 54.12, 42,... Measure time between arrivals Inter-arrival time 0, 1.3, 2.78,...

16 5 - 16 © 1998 Prentice-Hall, Inc. Continuous Probability Density Function 1.Mathematical formula 2.Shows all values, x, & frequencies, f(x) f(x) is not probability f(x) is not probability

17 5 - 17 © 1998 Prentice-Hall, Inc. Continuous Probability Density Function 1.Mathematical formula 2.Shows all values, x, & frequencies, f(x) f(x) is not probability f(x) is not probability Value (Value, Frequency) Frequency f(x) ab x

18 5 - 18 © 1998 Prentice-Hall, Inc. Continuous Probability Density Function 1.Mathematical formula 2.Shows all values, x, & frequencies, f(x) f(x) is not probability f(x) is not probability 3.Properties (Area under curve) Value (Value, Frequency) Frequency f(x) ab x fxdx fx () () All X a x b z   1 0,

19 5 - 19 © 1998 Prentice-Hall, Inc. Continuous Random Variable Probability

20 5 - 20 © 1998 Prentice-Hall, Inc. Continuous Random Variable Probability Probability is area under curve! © 1984-1994 T/Maker Co.

21 5 - 21 © 1998 Prentice-Hall, Inc. Continuous Random Variable Probability Probability is area under curve! © 1984-1994 T/Maker Co. Pcxdfxdx c d ()() z f(x) X cd

22 5 - 22 © 1998 Prentice-Hall, Inc. Normal Distribution

23 5 - 23 © 1998 Prentice-Hall, Inc. Continuous Probability Distribution Models

24 5 - 24 © 1998 Prentice-Hall, Inc. Importance of Normal Distribution 1.Describes many random processes or continuous phenomena 2.Can be used to approximate discrete probability distributions Example: binomial Example: binomial 3.Basis for classical statistical inference

25 5 - 25 © 1998 Prentice-Hall, Inc. Normal Distribution 1.‘Bell-shaped’ & symmetrical

26 5 - 26 © 1998 Prentice-Hall, Inc. Normal Distribution 1.‘Bell-shaped’ & symmetrical Mean Median Mode

27 5 - 27 © 1998 Prentice-Hall, Inc. 1.‘Bell-shaped’ & symmetrical Normal Distribution 2.Mean, median, mode are equal 3. Random variable has infinite range Mean Median Mode

28 5 - 28 © 1998 Prentice-Hall, Inc. Probability Density Function f(x)=Frequency of random variable x  =Population standard deviation  =3.14159; e = 2.71828 x=Value of random variable (-  < x <  )  =Population mean

29 5 - 29 © 1998 Prentice-Hall, Inc. Effect of Varying Parameters (  &  )

30 5 - 30 © 1998 Prentice-Hall, Inc. Effect of Varying Parameters (  &  ) X f(X)

31 5 - 31 © 1998 Prentice-Hall, Inc. Effect of Varying Parameters (  &  ) X f(X) A

32 5 - 32 © 1998 Prentice-Hall, Inc. Effect of Varying Parameters (  &  ) X f(X) A B

33 5 - 33 © 1998 Prentice-Hall, Inc. Effect of Varying Parameters (  &  )

34 5 - 34 © 1998 Prentice-Hall, Inc. Normal Distribution Probability

35 5 - 35 © 1998 Prentice-Hall, Inc. Normal Distribution Probability Probability is area under curve!

36 5 - 36 © 1998 Prentice-Hall, Inc. Normal Distribution Probability Probability is area under curve!

37 5 - 37 © 1998 Prentice-Hall, Inc. Normal Distribution Probability Probability is area under curve! I’ll use tables!

38 5 - 38 © 1998 Prentice-Hall, Inc. Normal Distribution Probability Tables Normal distributions differ by mean & standard deviation.

39 5 - 39 © 1998 Prentice-Hall, Inc. Normal Distribution Probability Tables Normal distributions differ by mean & standard deviation. Each distribution would require its own table. That’s an infinite number!

40 5 - 40 © 1998 Prentice-Hall, Inc. Standardize the Normal Distribution

41 5 - 41 © 1998 Prentice-Hall, Inc. Standardize the Normal Distribution Normal Distribution

42 5 - 42 © 1998 Prentice-Hall, Inc. Standardize the Normal Distribution One table! Normal Distribution Standardized Normal Distribution

43 5 - 43 © 1998 Prentice-Hall, Inc. Standardizing Example

44 5 - 44 © 1998 Prentice-Hall, Inc. Standardizing Example Normal Distribution

45 5 - 45 © 1998 Prentice-Hall, Inc. Standardizing Example Normal Distribution

46 5 - 46 © 1998 Prentice-Hall, Inc. Standardizing Example Normal Distribution Standardized Normal Distribution

47 5 - 47 © 1998 Prentice-Hall, Inc. Obtaining the Probability.0478.0478.02 0.1.0478 Standardized Normal Probability Table (Portion) ProbabilitiesProbabilities Shaded area exaggerated

48 5 - 48 © 1998 Prentice-Hall, Inc. Example P(3.8  X  5)

49 5 - 49 © 1998 Prentice-Hall, Inc. Example P(3.8  X  5) Normal Distribution

50 5 - 50 © 1998 Prentice-Hall, Inc. Example P(3.8  X  5) Normal Distribution

51 5 - 51 © 1998 Prentice-Hall, Inc. Example P(3.8  X  5) Normal Distribution.0478 Standardized Normal Distribution Shaded area exaggerated

52 5 - 52 © 1998 Prentice-Hall, Inc. Example P(2.9  X  7.1)

53 5 - 53 © 1998 Prentice-Hall, Inc. Example P(2.9  X  7.1) Normal Distribution

54 5 - 54 © 1998 Prentice-Hall, Inc. Example P(2.9  X  7.1) Normal Distribution Shaded area exaggerated

55 5 - 55 © 1998 Prentice-Hall, Inc. Example P(2.9  X  7.1) Normal Distribution.1664.1664.0832.0832 Standardized Normal Distribution Shaded area exaggerated

56 5 - 56 © 1998 Prentice-Hall, Inc. Example P( X  8)

57 5 - 57 © 1998 Prentice-Hall, Inc. Example P( X  8) Normal Distribution

58 5 - 58 © 1998 Prentice-Hall, Inc. Example P( X  8) Normal Distribution

59 5 - 59 © 1998 Prentice-Hall, Inc. Example P( X  8) Normal Distribution Standardized Normal Distribution.1179.1179.5000.3821.3821 Shaded area exaggerated

60 5 - 60 © 1998 Prentice-Hall, Inc. Example P(7.1  X  8)

61 5 - 61 © 1998 Prentice-Hall, Inc. Example P(7.1  X  8) Normal Distribution

62 5 - 62 © 1998 Prentice-Hall, Inc. Example P(7.1  X  8) Normal Distribution

63 5 - 63 © 1998 Prentice-Hall, Inc. Example P(7.1  X  8) Normal Distribution.0832.1179.0347.0347 Standardized Normal Distribution Shaded area exaggerated

64 5 - 64 © 1998 Prentice-Hall, Inc. Normal Distribution Thinking Challenge You work in Quality Control for GE. Light bulb life has a normal distribution with  = 2000 hours &  = 200 hours. What’s the probability that a bulb will last A. between 2000 & 2400 hours? B. less than 1470 hours? AloneGroupClass

65 5 - 65 © 1998 Prentice-Hall, Inc. Solution* P(2000  X  2400) Normal Distribution.4772.4772 Standardized Normal Distribution

66 5 - 66 © 1998 Prentice-Hall, Inc. Solution* P( X  1470) Normal Distribution.4960.4960.0040.0040.5000 Standardized Normal Distribution

67 5 - 67 © 1998 Prentice-Hall, Inc. Finding Z Values for Known Probabilities

68 5 - 68 © 1998 Prentice-Hall, Inc. Finding Z Values for Known Probabilities.1217.1217 What is Z given P(0 < Z < z0) =.1217? Shaded area exaggerated

69 5 - 69 © 1998 Prentice-Hall, Inc. Finding Z Values for Known Probabilities.1217.1217.01 0.3.1217 Standardized Normal Probability Table (Portion) What is Z given P(0 < Z < z0) =.1217? Shaded area exaggerated

70 5 - 70 © 1998 Prentice-Hall, Inc. Finding Z Values for Known Probabilities.1217.1217.01 0.3.1217 Standardized Normal Probability Table (Portion) What is Z given P(0 < Z < z0) =.1217? Shaded area exaggerated

71 5 - 71 © 1998 Prentice-Hall, Inc. Finding X Values for Known Probabilities

72 5 - 72 © 1998 Prentice-Hall, Inc. Finding X Values for Known Probabilities Normal Distribution.1217.1217 Shaded areas exaggerated

73 5 - 73 © 1998 Prentice-Hall, Inc. Finding X Values for Known Probabilities Normal Distribution Standardized Normal Distribution.1217.1217 Shaded areas exaggerated

74 5 - 74 © 1998 Prentice-Hall, Inc. Finding X Values for Known Probabilities Normal Distribution Standardized Normal Distribution.1217.1217 Shaded areas exaggerated

75 5 - 75 © 1998 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution

76 5 - 76 © 1998 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution 1.Not all binomial tables exist

77 5 - 77 © 1998 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution 1.Not all binomial tables exist 2.Use normal distrib. to approximate

78 5 - 78 © 1998 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution 1.Not all binomial tables exist 2.Use normal distrib. to approximate n = 10 p = 0.50.0.1.2.3 0246810 X P(X)

79 5 - 79 © 1998 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution 1.Not all binomial tables exist 2.Use normal distrib. to approximate 3.Requires large sample size n = 10 p = 0.50.0.1.2.3 0246810 X P(X)

80 5 - 80 © 1998 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution 1.Not all binomial tables exist 2.Use normal distrib. to approximate 3.Requires large sample size 4.Need correction for continuity n = 10 p = 0.50.0.1.2.3 0246810 X P(X)

81 5 - 81 © 1998 Prentice-Hall, Inc. Why Probability Is Approximate

82 5 - 82 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate

83 5 - 83 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate

84 5 - 84 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height

85 5 - 85 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height

86 5 - 86 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5

87 5 - 87 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5

88 5 - 88 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Probability added by normal curve Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5

89 5 - 89 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Probability added by normal curve Normal probability: Area under curve from 3.5 to 4.5

90 5 - 90 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Probability added by normal curve Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5

91 5 - 91 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Probability added by normal curve Normal probability: Area under curve from 3.5 to 4.5

92 5 - 92 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Probability added by normal curve Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5

93 5 - 93 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5 Probability added by normal curve Probability lost by normal curve

94 5 - 94 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Probability added by normal curve Normal probability: Area under curve from 3.5 to 4.5 Probability lost by normal curve

95 5 - 95 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5 Probability added by normal curve Probability lost by normal curve

96 5 - 96 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Probability added by normal curve Normal probability: Area under curve from 3.5 to 4.5 Probability lost by normal curve

97 5 - 97 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Why Probability Is Approximate Binomial probability: Bar height Normal probability: Area under curve from 3.5 to 4.5 Probability added by normal curve Probability lost by normal curve

98 5 - 98 © 1998 Prentice-Hall, Inc. Correction for Continuity 1.A 1/2 unit adjustment to discrete variable 2.Used when approximating a discrete distribution with a continuous distribution 3.Improves accuracy

99 5 - 99 © 1998 Prentice-Hall, Inc. Correction for Continuity 1.A 1/2 unit adjustment to discrete variable 2.Used when approximating a discrete distribution with a continuous distribution 3.Improves accuracy 4.5 (4 +.5) 3.5 (4 -.5) 4

100 5 - 100 © 1998 Prentice-Hall, Inc. Normal Approximation Procedure

101 5 - 101 © 1998 Prentice-Hall, Inc. Normal Approximation Procedure 1.Calculate the interval: If interval lies in range 0 to n, normal approximation can be used If interval lies in range 0 to n, normal approximation can be used

102 5 - 102 © 1998 Prentice-Hall, Inc. Normal Approximation Procedure 1.Calculate the interval: If interval lies in range 0 to n, normal approximation can be used If interval lies in range 0 to n, normal approximation can be used 2.Express binomial probability in form:

103 5 - 103 © 1998 Prentice-Hall, Inc. Normal Approximation Procedure 1.Calculate the interval: If interval lies in range 0 to n, normal approximation can be used If interval lies in range 0 to n, normal approximation can be used 2.Express binomial probability in form: 3.For each value of interest, a, use:

104 5 - 104 © 1998 Prentice-Hall, Inc..0.1.2.3 0246810 x P(x) Normal Approximation Example 3.54.5 What is the normal approximation of P(x = 4) given n = 10, and p = 0.5?

105 5 - 105 © 1998 Prentice-Hall, Inc. Normal Approximation Solution 1.Calculate the interval: Interval lies in range 0 to 10, so normal approximation can be used Interval lies in range 0 to 10, so normal approximation can be used

106 5 - 106 © 1998 Prentice-Hall, Inc. Normal Approximation Solution 1.Calculate the interval: Interval lies in range 0 to 10, so normal approximation can be used Interval lies in range 0 to 10, so normal approximation can be used 2.Express binomial probability in form:

107 5 - 107 © 1998 Prentice-Hall, Inc. Normal Approximation Solution Z (a +.5) np npp        ()..... 1 35105 10515 95 afaf afafaf Z np npp        ()..... 1 45105 10515 32 afaf afafaf (b +.5) 3.Compute standard normal z values:

108 5 - 108 © 1998 Prentice-Hall, Inc.  = 0 = 0  = 1 = 1 -.32 Z -.95 Normal Approximation Solution.1255.3289 -.1255.2034.3289 4.Sketch the approximate normal distribution:

109 5 - 109 © 1998 Prentice-Hall, Inc. Normal Approximation Solution.0.1.2.3 0246810 x P(x) 5.The exact probability from the binomial equation is 0.2000 (vs. 0.2034)

110 5 - 110 © 1998 Prentice-Hall, Inc. Conclusion 1.Defined continuous random variable 2.Described the normal random variables 3.Calculated probabilities for Normal random variables 4.Approximated the binomial distribution using the normal distribution

111 5 - 111 © 1998 Prentice-Hall, Inc. This Class... 1.What was the most important thing you learned in this chapter? 2.What do you still have questions about? 3.How can the lectures be improved? Please take a moment to answer the following questions in writing:

112 End of Chapter Any blank slides that follow are blank intentionally.

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