1. Continuous Probability Distributions
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 =
 = Population mean

11. Normal Probability Distribution (Figure 5-2)
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
0.52
Example:
z = 0.52 (or -0.52)
P(0 < z < .52) = 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

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

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 =
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

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