Continuous Probability Distributions

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.

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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

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Continuous Probability Distributions

• The probability distribution of a continuous
  random variable is represented by a probability
  density function that defines a curve.

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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
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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.

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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

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Normal Probability Distribution(Figure 5-2)
Probability 0.50
Probability 0.50
X
?
Mean Median Mode
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Difference Between Normal Distributions(Figure
5-3)
x
(a)
x
(b)
x
(c)
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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.

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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)

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Areas Under the Standard Normal Curve(Using
Table 5-1)
0.1985
X
0
0.52
Example z 0.52 (or -0.52) A(z) 0.1985 or
19.85
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Areas Under the Standard Normal Curve(Table 5-1)
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Standard Normal Example(Figure 5-6)
Probabilities from the Normal Curve for Westex
0.1915
0.50
x
z
50
x45
0
Z-.50
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Standard Normal Example(Figure 5-7)
z
z1.25
x7.5
From the normal table P(-1.25 ? z ? 0)
0.3944 Then, P(x ? 7.5 hours) 0.50 - 0.3944
0.1056
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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.

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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

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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.

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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 (? gt0)

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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)
Values of x
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Exponential Probability

• EXPONENTIAL PROBABILITY

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Key Terms

• Continuous Random Variable
• Discrete Random Variable
• Exponential Distribution

• Normal Distribution
• Standard Normal Distribution Standard Normal
  Table
• Uniform Distribution
• z-Value