1. 5-1

2. 5-2 Chapter Five Continuous Random Variables McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

3. 5-3 Continuous Random Variables 5.1 Continuous Probability Distributions 5.2 The Uniform Distribution 5.3 The Normal Probability Distribution *5.4Approximating the Binomial Distribution by Using the Normal Distribution *5.5The Exponential Distribution *5.6 The Cumulative Normal Table

4. 5-4 5.1 Continuous Probability Distributions The curve f(x) is the continuous probability distribution (or probability curve or probability density function) of the random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval. Properties of f(x) 1.f(x)  0 for all x 2.The total area under the curve of f(x) is equal to 1 20061116 start

5. 5-5 5.2 The Uniform Distribution If c and d are numbers on the real line, the probability curve describing the uniform distribution is The mean and standard deviation of a uniform random variable x are

6. 5-6 The Uniform Probability Curve

7. 5-7 5.3 The Normal Probability Distribution The normal probability distribution is defined by the equation  and  are the mean and standard deviation,  = 3.14159 … and e = 2.71828 is the base of natural or Naperian logarithms.

8. 5-8 The Position and Shape of the Normal Curve

9. 5-9 Normal Probabilities

10. 5-10 Three Important Areas under the Normal Curve The Empirical Rule for Normal Populations

11. 5-11 The Standard Normal Distribution If x is normally distributed with mean  and standard deviation , then is normally distributed with mean 0 and standard deviation 1, a standard normal distribution.

12. 5-12 Some Areas under the Standard Normal Curve

13. 5-13 Calculating P(z  -1)

14. 5-14 Calculating P(z  1)

15. 5-15 Finding Normal Probabilities Example 5.2 The Car Mileage Case Procedure 1.Formulate in terms of x. 2.Restate in terms of relevant z values. 3.Find the indicated area under the standard normal curve.

16. 5-16 Finding Z Points on a Standard Normal Curve

17. 5-17 Finding X Points on a Normal Curve Example 5.5 Finding the number of tapes stocked so that P(x > st) = 0.05

18. 5-18 Finding a Tolerance Interval Finding a tolerance interval [   k  ] that contains 99% of the measurements in a normal population.

19. 5-19 5.4 Normal Approximation to the Binomial If x is binomial, n trials each with probability of success p and n and p are such that np  5 and n(1-p)  5, then x is approximately normal with

20. 5-20 Example: Normal Approximation to Binomial Example 5.8: Approximating the binomial probability P(x = 23) by using the normal curve when Continuity correction: 查 z 值表

21. 5-21 5.5 The Exponential Distribution If  is positive  then the exponential distribution is described by the probability density function mean  x =1/ standard deviation  x =1/ 靠積分 (page 220)

22. 5-22 Example: Computing Exponential Probabilities Given  x =3.0 or =1/3=.333, xx 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 59 x =0.333

23. 5-23 5.6 The Cumulative Normal Table The cumulative normal table gives of being less than or equal any given z-value The cumulative normal table gives the shaded area

24. 5-24 Discrete Random Variables 5.1 Continuous Probability Distributions 5.2 The Uniform Distribution 5.3 The Normal Probability Distribution *5.4Approximating the Binomial Distribution by Using the Normal Distribution *5.5The Exponential Distribution *5.6 The Cumulative Normal Table Summary: