bowerman_9e_chap_071.pptx

Chapter 7
Continuous Random Variables

Copyright ©2018 McGraw-Hill Education. All rights reserved.

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Chapter Outline
7.1 Continuous Probability Distributions
7.2 The Uniform Distribution
7.3 The Normal Probability Distribution
7.4 Approximating the Binomial Distribution by Using the Normal Distribution (Optional)
7.5 The Exponential Distribution (Optional)
7.6 The Normal Probability Plot (Optional)
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7.1 Continuous Probability Distributions
A continuous random variable may assume any numerical value in one or more intervals
Car mileage
Temperature

Use a continuous probability distribution to assign probabilities to intervals of values
LO7-1: Define a continuous probability distribution and explain how it is used.
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Continuous Probability Distributions Continued
The curve f(x) is the continuous probability distribution of the continuous 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

Other names for a continuous probability distribution are probability curve and probability density function

We will look at the uniform, normal, and exponential distributions

LO7-1
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Properties of Continuous Probability Distributions
Properties of f(x): f(x) is a continuous function such that
f(x) ≥ 0 for all x
The total area under the curve of f(x) is equal to 1

Essential point: An area under a continuous probability distribution is a probability
LO7-1
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7.2 The Uniform Distribution
LO7-2: Use the uniform distribution to compute probabilities.
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The Uniform Distribution Mean and Standard Deviation
LO7-2
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LO7-2
The Uniform Probability Curve
Figure 7.2 (b)

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Example 7.1 Elevator Waiting Time
Elevator wait time
Uniform 0 – 4
c = 0
d = 4
LO7-2
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7.3 The Normal Probability Distribution

π = 3.14159
e = 2.71828
LO7-3: Describe the properties of the normal distribution and use a cumulative normal table.
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LO7-3
The Normal Probability Distribution Continued
Figure 7.3

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Properties of the Normal Distribution
There are an infinite number of normal curves
The shape of any individual normal curve depends on its specific mean and standard deviation
The highest point is over the mean
Also the median and mode
The curve is symmetrical about its mean
The left and right halves of the curve are mirror images of each other
LO7-3
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Properties of the Normal Distribution Continued
The tails of the normal extend to infinity in both directions
The tails get closer to the horizontal axis but never touch it

The area under the normal curve to the right of the mean equals the area under the normal curve to the left of the mean
The area under each half is 0.5
LO7-3
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LO7-3
The Position and Shape of the Normal Curve
Figure 7.4

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LO7-3
Normal Probabilities
Figure 7.5

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LO7-3
Three Important Percentages
Figure 7.6

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LO7-3
Finding Normal Curve Areas
Figure 7.7

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LO7-3
The Cumulative Normal Table
Top of Table 7.1

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z = -2.33, probability = 0.0099

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LO7-3
Examples
Figures 7.8 and 7.9

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LO7-3
Examples Continued
Figures 7.10 and 7.11

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LO7-3
Examples Continued
Figures 7.12 and 7.13

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Finding Normal Probabilities
Formulate the problem in terms of x values
Calculate the corresponding z values, and restate the problem in terms of these z values

Find the required areas under the standard normal curve by using the table

Note: It is always useful to draw a picture showing the required areas before using the normal table
LO7-4: Use the normal distribution to compute probabilities.
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Finding a Point on the Horizontal Axis Under a Normal Curve
Figure 7.19
LO7-5: Find population values that correspond to specified normal distribution probabilities.

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7.4 Approximating the Binomial Distribution by Using the Normal Distribution (Optional)
Suppose x is a binomial random variable
n is the number of trials
Each having a probability of success p

If np  5 and nq  5, then x is approximately normal with a mean of np and a standard deviation of the square root of npq
LO7-6: Use the normal distribution to approximate binomial probabilities (Optional).
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LO7-6
Approximating the Binomial Probability Using the Normal Curve
Figure 7.23

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7.5 The Exponential Distribution (Optional)
Suppose that some event occurs as a Poisson process
That is, the number of times an event occurs is a Poisson random variable

Let x be the random variable of the interval between successive occurrences of the event
The interval can be some unit of time or space

Then x is described by the exponential distribution
With parameter λ, which is the mean number of events that can occur per given interval
LO7-7: Use the exponential distribution to compute probabilities (Optional).
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The Exponential Distribution Continued
LO7-7
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LO7-7
The Exponential Distribution Continued
Figure 7.25

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Example 7.9 The Air Safety Case: Traffic Control Errors
λ = 20.8 errors per year
λ = 0.4 errors per week
Probability of one to two weeks
LO7-7
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7.6 The Normal Probability Plot (Optional)
A graphic used to visually check to see if sample data comes from a normal distribution

A straight line indicates a normal distribution

The more curved the line, the less normal the data is distributed
LO7-8: Use a normal probability plot to help decide whether data come from a normal distribution (Optional).
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Creating a Normal Probability Plot
Rank the data from smallest to largest
For each data point, compute the value
/(n + 1)
is the data point’s position in the list
For each data point, compute the standardized normal quantile value (O)
O is the z value that gives an area /(n + 1) to its left
Plot data points against O
Straight line indicates normal distribution
LO7-8
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LO7-8
Sample Normal Probability Plots
Figures 7.27, 7.28 and 7.29

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