5-1

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

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

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 start

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

5-6 The Uniform Probability Curve

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

5-8 The Position and Shape of the Normal Curve

5-9 Normal Probabilities

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

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.

5-12 Some Areas under the Standard Normal Curve

5-13 Calculating P(z  -1)

5-14 Calculating P(z  1)

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.

5-16 Finding Z Points on a Standard Normal Curve

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

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

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

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

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)

5-22 Example: Computing Exponential Probabilities Given  x =3.0 or =1/3=.333, xx x =0.333

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

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: