1 / 26

Chapter 5

Chapter 5. Continuous Probability Distributions. Chapter 5 - Chapter Outcomes. After studying the material in this chapter, you should be able to: • Discuss the important properties of the normal probability distribution.

taite
Download Presentation

Chapter 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 5 Continuous Probability Distributions

  2. Chapter 5 - Chapter Outcomes After studying the material in this chapter, you should be able to: • Discuss the important properties of the normal probability distribution. • Recognize when the normal distribution might apply in a decision-making process.

  3. Chapter 5 - Chapter Outcomes(continued) After studying the material in this chapter, you should be able to: • Calculate probabilities using the normal distribution table and be able to apply the normal distribution in appropriate business situations. • Recognize situations in which the uniform and exponential distributions apply.

  4. Continuous Probability Distributions A discrete random variable is a variable that can take on a countable number of possible values along a specified interval.

  5. Continuous Probability Distributions A continuous random variable is a variable that can take on any of the possible values between two points.

  6. Examples of Continuous Random variables • Time required to perform a job • Financial ratios • Product weights • Volume of soft drink in a 12-ounce can • Interest rates • Income levels • Distance between two points

  7. Continuous Probability Distributions The probability distribution of a continuous random variable is represented by a probability density function that defines a curve.

  8. Continuous Probability Distributions (a) Discrete Probability Distribution (b) Probability Density Function P(x) f(x) x x Possible Values of x Possible Values of x

  9. Normal Probability Distribution The Normal Distribution is a bell-shaped, continuous distribution with the following properties: 1. It is unimodal. 2. It is symmetrical; this means 50% of the area under the curve lies left of the center and 50% lies right of center. 3. The mean, median, and mode are equal. 4. It is asymptotic to the x-axis. 5. The amount of variation in the random variable determines the width of the normal distribution.

  10. Normal Probability Distribution NORMAL DISTRIBUTION DENSITY FUNCTION where: x = Any value of the continuous random variable  = Population standard deviation e = Base of the natural log = 2.7183  = Population mean

  11. Normal Probability Distribution(Figure 5-2) Probability = 0.50 Probability = 0.50 f(x) x  Mean Median Mode

  12. Differences Between Normal Distributions(Figure 5-3) x (a) x (b) x (c)

  13. Standard Normal Distribution The standard normal distribution is a normal distribution which has a mean = 0.0 and a standard deviation = 1.0. The horizontal axis is scaled in standardized z-values that measure the number of standard deviations a point is from the mean. Values above the mean have positive z-values and those below have negative z-values.

  14. Standard Normal Distribution STANDARDIZED NORMAL Z-VALUE where: x = Any point on the horizontal axis  = Standard deviation of the normal distribution  = Population mean z = Scaled value (the number of standard deviations a point x is from the mean)

  15. Areas Under the Standard Normal Curve(Using Table 5-1) 0.1985 X 0 0.52 Example: z = 0.52 (or -0.52) P(0 < z < .52) = 0.1985 or 19.85%

  16. Areas Under the Standard Normal Curve(Table 5-1)

  17. Standard Normal Example(Figure 5-6) Probabilities from the Normal Curve for Westex 0.1915 0.50 x z x=45 50 z= -.50 0

  18. Standard Normal Example(Figure 5-7) z z=-1.25 x=7.5 From the normal table: P(-1.25  z  0) = 0.3944 Then, P(x  7.5 hours) = 0.50 - 0.3944 = 0.1056

  19. Uniform Probability Distribution The uniform distribution is a probability distribution in which the probability of a value occurring between two points, a and b, is the same as the probability between any other two points, c and d, given that the distribution between a and b is equal to the distance between c and d.

  20. Uniform Probability Distribution CONTINUOUS UNIFORM DISTRIBUTION where: f(x) = Value of the density function at any x value a = Lower limit of the interval from a to b b = Upper limit of the interval from a to b

  21. Uniform Probability Distributions(Figure 5-16) f(x) f(x) for 2  x  5 for 3  x  8 .50 .50 .25 .25 2 5 3 8 a b a b

  22. Exponential Probability Distribution The exponential probability distribution is a continuous distribution that is used to measure the time that elapses between two occurrences of an event.

  23. Exponential Probability Distribution EXPONENTIAL DISTRIBUTION A continuous random variable that is exponentially distributed has the probability density function given by: where: e = 2.71828. . . 1/ = The mean time between events ( >0)

  24. Exponential Distributions(Figure 5-18) Lambda = 3.0 (Mean = 0.333) f(x) Lambda = 2.0 (Mean = 0.5) Lambda = 1.0 (Mean = 1.0) Lambda = 0.50 (Mean = 020) x Values of x

  25. Exponential Probability EXPONENTIAL PROBABILITY

  26. • Continuous Random Variable • Discrete Random Variable • Exponential Distribution • Normal Distribution • Standard Normal Distribution Standard Normal Table • Uniform Distribution • z-Value Key Terms

More Related