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

7-4

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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 order 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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