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

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

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

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

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

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

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

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

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

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

© 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,...

© 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

© 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,...

© 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

© 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,...

© 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

© 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

© 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,

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

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

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

© 1998 Prentice-Hall, Inc. Normal Distribution

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

© 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

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

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

© 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

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

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

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

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

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

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

© 1998 Prentice-Hall, Inc. Normal Distribution Probability

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

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

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

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

© 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!

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

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

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

© 1998 Prentice-Hall, Inc. Standardizing Example

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

© 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

© 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 = X P(X)

© 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 = X P(X)

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

© 1998 Prentice-Hall, Inc x P(x) Why Probability Is Approximate

© 1998 Prentice-Hall, Inc x P(x) Why Probability Is Approximate

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

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

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

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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 1998 Prentice-Hall, Inc 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

© 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

© 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

© 1998 Prentice-Hall, Inc. Normal Approximation Procedure

© 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

© 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:

© 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:

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

© 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

© 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:

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

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

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

© 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

© 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:

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