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

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

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

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. The Uniform Probability Curve

7. 5.3 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.

8. The Position and Shape of the Normal Curve

9. Normal Probabilities

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

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. Some Areas under the Standard Normal Curve

13. Calculating P(z  -1)

14. Calculating P(z  1)

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

16. Finding Z Points on a Standard Normal Curve

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

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. Example: Normal Approximation to Binomial
Example 5.8: Approximating the binomial probability P(x = 23) by using the normal curve when

21. 5.5 The Exponential Distribution
If l is positive, then the exponential distribution is described by the probability density function
mean mx=1/l
standard deviation sx=1/l

22. Example: Computing Exponential Probabilities
Given mx=3.0 or l=1/3=.333,
0.05
0.1
0.15
0.2
0.25
0.3
0.35 5 9
l=0.333 x mx

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