5 - 1 © 2000 Prentice-Hall, Inc. Statistics for Business and Economics Continuous Random Variables Chapter 5

5 - 2 © 2000 Prentice-Hall, Inc. Learning Objectives 1.Define Continuous Random Variable 2.Describe the Uniform, Normal, & Exponential Random Variables 3.Calculate Probabilities for Continuous Random Variables 4.Approximate the Binomial Distribution Using the Normal Distribution

5 - 3 © 2000 Prentice-Hall, Inc. Data Types

5 - 4 © 2000 Prentice-Hall, Inc. Continuous Random Variables

5 - 5 © 2000 Prentice-Hall, Inc. Continuous Random Variable 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 © 2000 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,...

5 - 7 © 2000 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    1 0,

5 - 8 © 2000 Prentice-Hall, Inc. Continuous Random Variable Probability Probability Is Area Under Curve! © T/Maker Co. Pcxdfxdx c d ()()  f(x) X cd

5 - 9 © 2000 Prentice-Hall, Inc. Continuous Probability Distribution Models

© 2000 Prentice-Hall, Inc.

© 2000 Prentice-Hall, Inc. Uniform Distribution

© 2000 Prentice-Hall, Inc. Continuous Probability Distribution Models

© 2000 Prentice-Hall, Inc. Uniform Distribution 1. Equally Likely Outcomes 2. Probability Density 3. Mean & Standard Deviation Mean Median x f(x)f(x) dc

© 2000 Prentice-Hall, Inc. Uniform Distribution Example You’re production manager of a soft drink bottling company. You believe that when a machine is set to dispense., it really dispenses 11.5 to 12.5 oz. inclusive. Suppose the amount dispensed has a uniform distribution. What is the probabiilty that less than 11.8 oz. is dispensed? You’re production manager of a soft drink bottling company. You believe that when a machine is set to dispense 12 oz., it really dispenses 11.5 to 12.5 oz. inclusive. Suppose the amount dispensed has a uniform distribution. What is the probabiilty that less than 11.8 oz. is dispensed? SODA

© 2000 Prentice-Hall, Inc. Uniform Distribution Solution P(11.5  x  11.8)= (Base)(Height) = ( )(1) = f(x)f(x)f(x)f(x) x

© 2000 Prentice-Hall, Inc. Normal Distribution

© 2000 Prentice-Hall, Inc. Continuous Probability Distribution Models

© 2000 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

© 2000 Prentice-Hall, Inc. Normal Distribution 1.‘Bell-Shaped’ & Symmetrical 2.Mean, Median, Mode Are Equal 3.‘Middle Spread’ Is 1.33  4. Random Variable Has Infinite Range Mean Median Mode

© 2000 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

© 2000 Prentice-Hall, Inc. Effect of Varying Parameters (  &  )

© 2000 Prentice-Hall, Inc. Normal Distribution Probability Probability is area under curve!

© 2000 Prentice-Hall, Inc. Infinite Number of Tables Normal distributions differ by mean & standard deviation.

© 2000 Prentice-Hall, Inc. Infinite Number of Tables Normal distributions differ by mean & standard deviation. Each distribution would require its own table. That’s an infinite number!

© 2000 Prentice-Hall, Inc. Standardize the Normal Distribution

© 2000 Prentice-Hall, Inc. Standardize the Normal Distribution Normal Distribution

© 2000 Prentice-Hall, Inc. Standardize the Normal Distribution One table! Normal Distribution Standardized Normal Distribution

© 2000 Prentice-Hall, Inc. Standardizing Example

© 2000 Prentice-Hall, Inc. Standardizing Example Normal Distribution

© 2000 Prentice-Hall, Inc. Standardizing Example Normal Distribution

© 2000 Prentice-Hall, Inc. Standardizing Example Normal Distribution Standardized Normal Distribution

© 2000 Prentice-Hall, Inc. Obtaining the Probability Standardized Normal Probability Table (Portion) ProbabilitiesProbabilities Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Example P(3.8  X  5)

© 2000 Prentice-Hall, Inc. Example P(3.8  X  5) Normal Distribution

© 2000 Prentice-Hall, Inc. Example P(3.8  X  5) Normal Distribution

© 2000 Prentice-Hall, Inc. Example P(3.8  X  5) Normal Distribution.0478 Standardized Normal Distribution Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Example P(2.9  X  7.1)

© 2000 Prentice-Hall, Inc. Example P(2.9  X  7.1) Normal Distribution

© 2000 Prentice-Hall, Inc. Example P(2.9  X  7.1) Normal Distribution Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Example P(2.9  X  7.1) Normal Distribution Standardized Normal Distribution Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Example P( X  8)

© 2000 Prentice-Hall, Inc. Example P( X  8) Normal Distribution

© 2000 Prentice-Hall, Inc. Example P( X  8) Normal Distribution

© 2000 Prentice-Hall, Inc. Example P( X  8) Normal Distribution Standardized Normal Distribution Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Example P(7.1  X  8)

© 2000 Prentice-Hall, Inc. Example P(7.1  X  8) Normal Distribution

© 2000 Prentice-Hall, Inc. Example P(7.1  X  8) Normal Distribution

© 2000 Prentice-Hall, Inc. Example P(7.1  X  8) Normal Distribution Standardized Normal Distribution Shaded area exaggerated

© 2000 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?

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

© 2000 Prentice-Hall, Inc. Solution* P( X  1470) Normal Distribution Standardized Normal Distribution

© 2000 Prentice-Hall, Inc. Finding Z Values for Known Probabilities

© 2000 Prentice-Hall, Inc. Finding Z Values for Known Probabilities What is Z given P(Z) =.1217? Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Finding Z Values for Known Probabilities Standardized Normal Probability Table (Portion) What is Z given P(Z) =.1217? Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Finding Z Values for Known Probabilities Standardized Normal Probability Table (Portion) What is Z given P(Z) =.1217? Shaded area exaggerated

© 2000 Prentice-Hall, Inc. Finding X Values for Known Probabilities

© 2000 Prentice-Hall, Inc. Finding X Values for Known Probabilities Normal Distribution Shaded areas exaggerated

© 2000 Prentice-Hall, Inc. Finding X Values for Known Probabilities Normal Distribution Standardized Normal Distribution Shaded areas exaggerated

© 2000 Prentice-Hall, Inc. Finding X Values for Known Probabilities Normal Distribution Standardized Normal Distribution Shaded areas exaggerated

© 2000 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution

© 2000 Prentice-Hall, Inc. Normal Approximation of Binomial Distribution 1.Not All Binomial Tables Exist 2.Requires Large Sample Size 3.Gives Approximate Probability Only 4.Need Correction for Continuity n = 10 p = X P(X)

© 2000 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

© 2000 Prentice-Hall, Inc x P(x) Why Probability Is Approximate Probability Added by Normal Curve Probability Lost by Normal Curve Binomial Probability: Bar Height Normal Probability: Area Under Curve from 3.5 to 4.5

© 2000 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

© 2000 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:

© 2000 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?

© 2000 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

© 2000 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:

© 2000 Prentice-Hall, Inc. Normal Approximation Solution Z (a +.5) np npp     () (.5) 10(.5)(1 -.5) 95 Z np npp     () (.5) 10(.5)(1 -.5) 32 (b +.5) 3.Compute Standard Normal z Values:

© 2000 Prentice-Hall, Inc.  = 0 = 0  = 1 = Z -.95 Normal Approximation Solution Sketch the Approximate Normal Distribution:

© 2000 Prentice-Hall, Inc. Normal Approximation Solution x P(x) 5.The Exact Probability From the Binomial Equation Is (vs )

© 2000 Prentice-Hall, Inc. Exponential Distribution

© 2000 Prentice-Hall, Inc. Continuous Probability Distribution Models

© 2000 Prentice-Hall, Inc. Exponential Distribution 1.Describes Time or Distance Between Events Used for Queues Used for Queues 2.Density Function 3.Parameters = 0.5 = 0.5 = 2.0 = 2.0

© 2000 Prentice-Hall, Inc. Exponential Distribution Probability

© 2000 Prentice-Hall, Inc. Exponential Distribution Example A college has a single counter at the Registrar’s Office. A student arrives on average every 10 minutes following an exponential distribution. What is the probability that more than 30 minutes pass without a student’s arriving? © 1995 Corel Corp.

© 2000 Prentice-Hall, Inc. Exponential Distribution Solution © 1995 Corel Corp.

© 2000 Prentice-Hall, Inc. Conclusion 1.Defined Continuous Random Variable 2.Described the Uniform, Normal, & Exponential Random Variables 3.Calculated Probabilities for Continuous Random Variables 4.Approximated the Binomial Distribution Using the Normal Distribution

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