chapter 7 additional integration topics section 2 applications in business and economics

Chapter 7

Additional Integration Topics

Section 2

Applications in Business and Economics

2Barnett/Ziegler/Byleen Business Calculus 12e

Learning Objectives for Section 7.2 Applications in Business/Economics

The student will be able to:

1. Construct and interpret probability density functions.

2. Evaluate a continuous income stream.

3. Evaluate the future value of a continuous income stream.

4. Evaluate consumers’ and producers’ surplus.

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Random Variables

Random variables come in two varieties: Discrete

• Values are distinct or separate and can be counted and listed

Continuous• Infinite number of values that are within an interval

Barnett/Ziegler/Byleen Business Calculus 12e

4

Continuous vs Discrete

Discrete Random Variable• Suppose we roll a single die. How many possible

outcomes are there?• There are 6 discrete possible outcomes.

Continuous Random Variable• Suppose we randomly choose a real number (x) in the

interval [1, 6]. How many possible outcomes are there?• There are an infinite number of possible outcomes.


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Continuous Random Variables

Suppose an experiment is designed in such a way that any real number x on the interval [c, d] is a possible outcome.

Examples of what x could represent:• Inches of rain in one day• Height of a person between 5 ft and 7 ft• Life of a lightbulb between 40 hours and 100 hours

These are all examples of continuous random variables because the possible outcomes are not discrete. Rather, there is an infinite number of possible outcomes over a specified interval.


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Probability Density Function

In certain situations, we can find a function that can be used to model the probability that a continuous random variable will take on a value over a specified interval.

Such functions are called probability density functions.



Probability Density Functions

A probability density function must satisfy 3 conditions:

1. f (x) 0 for all real x

2. The area under the graph of f (x) over the interval (-, ) is 1

3. If [c, d] is a subinterval of (-, ) then

the probability that x falls in the interval [c, d] is equal to:

d

cdxxf )(


Graph Examples

𝐴𝑟𝑒𝑎=1𝑐 𝑑


Example 1

In a certain city, the daily use of water in hundreds of gallons per household is a continuous random variable with probability density function

Find the probability that a household chosen at random will use between 300 and 600 gallons. (Use graphing calculator.)

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (3 ≤𝑥 ≤ 6 )=¿3

6

.15𝑒−.15 𝑥𝑑𝑥

≈ 0.23There is a 23% probability that a household chosen at random uses between 300-600 gallons of water.

f ( x )={.15𝑒− .15𝑥 𝑥≥ 00 h𝑜𝑡 𝑒𝑟𝑤𝑖𝑠𝑒


Conceptual Insight

The probability that a household in the previous example uses exactly 300 gallons is given by:

In fact, for any continuous random variable x with probability density function f (x), the probability that x is exactly equal to a constant c is equal to 0.

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (3 ≤𝑥 ≤ 3 )=¿3

3

.15𝑒−.15 𝑥𝑑𝑥

¿0

See khanAcademy.org “probability density functions” for an additional explanation.

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Example 2

Suppose that the length of a phone call (in minutes) is a continuous random variable with the probability density function:

Find the probability that a call selected at random will last 4 minutes or less. (Use graphing calculator.)

Solve for b so that the probability of a call selected at random lasting b minutes or less is 90%.


𝑓 (𝑡 )={0 .25𝑒− .25𝑡 𝑖𝑓 𝑡≥ 00 h𝑜𝑡 𝑒𝑟𝑤𝑖𝑠𝑒

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Example 2 (continued)


𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 (0 ≤ 𝑡≤ 4 )=¿0

4

.25𝑒−.25 𝑡𝑑𝑡

≈ 0.63 (𝑢𝑠𝑒𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑜𝑟 )There is a 63% probability that a phone call chosen at random will last 4 minutes or less.

Find the probability that a call selected at random will last 4 minutes or less.

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Solve for b so that the probability of a call selected at random lasting b minutes or less is 90%.

0

𝑏

.25𝑒−.25 𝑡𝑑𝑡=0.9

𝑢=− .25 𝑡𝑑𝑢=− .25𝑑𝑡

0

−.25 𝑏

−𝑒𝑢𝑑𝑢=0.9

−𝑒𝑢|− .25𝑏0

=0.9

−𝑒−.25 𝑏− (−𝑒0 )=0.9

−𝑒−.25 𝑏+1=0.9−𝑒−.25 𝑏=− 0. 1𝑒−.25𝑏=0.1

ln𝑒−.25 𝑏=ln 0.1− .25𝑏=ln 0.1𝑏≈ 9.21There is a 90% probability of a call lasting 9.21 minutes or less.

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Application: Continuous Income

A function that models the flow of money represents a continuous income stream.

Let f(t) represent the rate of flow of a continuous income stream where t is time.

We can use calculus to find the total income produced over a specified time interval.



Continuous Income Stream

Total Income for a Continuous Income Stream:

If f (t) is the rate of flow of a continuous income stream, the total income produced during the time period from t = a to t = b is

b

adttf )( income Total

a Total Income b

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Continuous Income Stream

This makes sense if you recall what we have been saying about definite integrals.

If you integrate a rate of change of a quantity on an interval then you get the total change of the quantity on that interval.

Since the rate of flow represents the rate of change of income produced then the definite integral from a to b represents the total income produced on that interval.



Example 3

Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is

f (t) = 600 e 0.06t

𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑐𝑜𝑚𝑒=0

2

600𝑒.06𝑡 𝑑𝑡

𝑢=.06 𝑡𝑑𝑢=.06𝑑𝑡 ¿

0

.12

10000 𝑒𝑢𝑑𝑢

¿100000

.12

𝑒𝑢𝑑𝑢10000𝑑𝑢=600𝑑𝑡

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¿10000 ∙𝑒𝑢|.120

¿10000 ∙(𝑒¿¿ .12−𝑒0)¿

¿10000 ∙(𝑒¿¿ .12 −1)¿

≈ 1 ,274.97

¿100000

.12

𝑒𝑢𝑑𝑢

The total income after the first 2 years is $1,274.97

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What would be the total income produced during the second two years? (Use graphing calculator.)

Interval will be [2, 4] because it represents the end of the 2nd year to the end of the 4th year.


𝑇𝑜𝑡𝑎𝑙𝑖𝑛𝑐𝑜𝑚𝑒=2

4

600𝑒.06𝑡 𝑑𝑡

≈ 1 ,437.52The total income produced during the next

two years is $1437.52

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Homework

#7-2APg 430

(13-19 odd, 21, 25)


khanAcademy.org “discrete and continuous random variables”“probability density functions”

chapter 7 additional integration topics section 2 applications in business and economics

Documents

e slide

continuous random variable

probability density

random uses

real x

real number x

economics slide

discrete possible outcomes