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2. Continuous Distributions Chapter 77 Continuous Variables Continuous Variables Describing a Continuous Distribution Describing a Continuous Distribution Uniform Continuous Distribution Uniform Continuous Distribution Normal Distribution Normal Distribution Standard Normal Distribution Standard Normal Distribution Normal Approximation to the Binomial (Optional) Normal Approximation to the Binomial (Optional) Normal Approximation to the Poisson (Optional) Normal Approximation to the Poisson (Optional) Exponential Distribution Exponential Distribution McGraw-Hill/Irwin© 2008 The McGraw-Hill Companies, Inc. All rights reserved.

3. 7-3 Continuous Variables Discrete Variable – each value of X has its own probability P(X).Discrete Variable – each value of X has its own probability P(X). Continuous Variable – events are intervals and probabilities are areas underneath smooth curves. A single point has no probability.Continuous Variable – events are intervals and probabilities are areas underneath smooth curves. A single point has no probability.  Events as Intervals

4. 7-4 Describing a Continuous Distribution Probability Density Function (PDF) – For a continuous random variable, the PDF is an equation that shows the height of the curve f (x) at each possible value of X over the range of X.Probability Density Function (PDF) – For a continuous random variable, the PDF is an equation that shows the height of the curve f (x) at each possible value of X over the range of X.  PDFs and CDFs Normal PDF

5. 7-5 Describing a Continuous Distribution Continuous PDF’s: Continuous PDF’s: Denoted f (x)Denoted f (x) Must be nonnegativeMust be nonnegative Total area under curve = 1Total area under curve = 1 Mean, variance and shape depend on the PDF parametersMean, variance and shape depend on the PDF parameters Reveals the shape of the distributionReveals the shape of the distribution  PDFs and CDFs Normal PDF

6. 7-6 Describing a Continuous Distribution Continuous CDF’s: Continuous CDF’s: Denoted F (x)Denoted F (x) Shows P(X < x), the cumulative proportion of scoresShows P(X < x), the cumulative proportion of scores Useful for finding probabilitiesUseful for finding probabilities  PDFs and CDFs Normal CDF

7. 7-7 Describing a Continuous Distribution Continuous probability functions are smooth curves. Continuous probability functions are smooth curves. Unlike discrete distributions, the area at any single point = 0.Unlike discrete distributions, the area at any single point = 0. The entire area under any PDF must be 1.The entire area under any PDF must be 1. Mean is the balance point of the distribution.Mean is the balance point of the distribution.  Probabilities as Areas

8. 7-8 Describing a Continuous Distribution  Expected Value and Variance

9. 7-9 Uniform Continuous Distribution  Characteristics of the Uniform Distribution

10. 7-10 Uniform Continuous Distribution  Example: Anesthesia Effectiveness An oral surgeon injects a painkiller prior to extracting a tooth. Given the varying characteristics of patients, the dentist views the time for anesthesia effectiveness as a uniform random variable that takes between 15 minutes and 30 minutes.An oral surgeon injects a painkiller prior to extracting a tooth. Given the varying characteristics of patients, the dentist views the time for anesthesia effectiveness as a uniform random variable that takes between 15 minutes and 30 minutes. X is U(15, 30)X is U(15, 30) a = 15, b = 30, find the mean and standard deviation.a = 15, b = 30, find the mean and standard deviation.

11. 7-11 Uniform Continuous Distribution  Example: Anesthesia Effectiveness  = = = = a + b 2 = 15 + 30 2 = 22.5 minutes  = = = = (b – a) 2 12 = 4.33 minutes (30 – 15) 2 12 = Find the probability that the anesthetic takes between 20 and 25 minutes. P(c < X < d) = (d – c)/(b – a) (25 – 20)/(30 – 15) P(20 < X < 25) = = 5/15 = 0.3333 or 33.33%

12. 7-12 Normal Distribution  Characteristics of the Normal Distribution Normal or Gaussian distribution was named for German mathematician Karl Gauss (1777 – 1855).Normal or Gaussian distribution was named for German mathematician Karl Gauss (1777 – 1855). Defined by two parameters,  and Defined by two parameters,  and  Denoted N( ,  )Denoted N( ,  ) Domain is –  < X < + Domain is –  < X < +  Almost all area under the normal curve is included in the range  – 3  < X <  + 3 Almost all area under the normal curve is included in the range  – 3  < X <  + 3 

13. 7-13 Normal Distribution  Characteristics of the Normal Distribution

14. 7-14 Normal Distribution  What is Normal? A normal random variable should: Be measured on a continuous scale.Be measured on a continuous scale. Possess clear central tendency.Possess clear central tendency. Have only one peak (unimodal).Have only one peak (unimodal). Exhibit tapering tails.Exhibit tapering tails. Be symmetric about the mean (equal tails).Be symmetric about the mean (equal tails).

15. 7-15 Standard Normal Distribution  Characteristics of the Standard Normal Since for every value of  and , there is a different normal distribution, we transform a normal random variable to a standard normal distribution with  = 0 and  = 1 using the formula:Since for every value of  and , there is a different normal distribution, we transform a normal random variable to a standard normal distribution with  = 0 and  = 1 using the formula: z =z =z =z = x –   Denoted N(0,1)Denoted N(0,1)

16. 7-16 Standard Normal Distribution  Characteristics of the Standard Normal

17. 7-17 Standard Normal Distribution  Finding Areas by using Standardized Variables Suppose John took an economics exam and scored 86 points. The class mean was 75 with a standard deviation of 7. What percentile is John in (i.e., find P(X < 86)?Suppose John took an economics exam and scored 86 points. The class mean was 75 with a standard deviation of 7. What percentile is John in (i.e., find P(X < 86)? z John = x –   = 86 – 75 7 = 11/7 = 1.57 So John’s score is 1.57 standard deviations about the mean.So John’s score is 1.57 standard deviations about the mean.