Learning Objectives for Section 14.2
Applications in Business/Economics
1. The student will be able to construct
and interpret probability density
functions.
2, The student will be able to evaluate a
continuous income stream.
3. The student will be able to evaluate
consumers’ and producers’ surplus.
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Probability Density Functions
A probability density function must satisfy:
1. f (x)  0 for all x
2. The area under the graph of f (x) is 1
3. If [c, d] is a subinterval then
Probability (c  x  d) =
Barnett/Ziegler/Byleen Business Calculus 11e

d
c
f ( x) dx
2
Probability Density Functions
(continued)
Sample probability density function
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Example
In a certain city, the daily use of water in hundreds of gallons
per household is a continuous random variable with
probability density function
f ( x)  .15e .15 x if x  0. Otherwise f ( x)  0
Find the probability that a household chosen at random will
use between 300 and 600 gallons.
6
P robability (3  x  6)   .15e  0.15 x dx
3
 e
 .15 x 6
|
3
 - e– 0 .9  e- 0 .45  0.23
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Insight
The probability that a household in the previous example uses
exactly 300 gallons is given by:
3
Probability (3  x  3)   .15e  0.15 x dx  0
3
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.
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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
T otalincome  f (t ) dt
a
a
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Total Income
b
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Example
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
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Example
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
2
T otalincome  600e 0.06 t dt
0
 10,000 e0.06t
2
0
 10,000 (e0.12 – 1 )
 $1,275
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Future Value
of a Continuous Income Stream
From previous work we are familiar with the continuous
compound interest formula
A = Pert.
If f (t) is the rate of flow of a continuous income stream,
0  t  T, and if the income is continuously invested at a rate r
compounded continuously, the the future value FV at the end of
T years is given by
FV 

T
0
f (t ) e
r (T  t )
Barnett/Ziegler/Byleen Business Calculus 11e
dt  e
rT

T
0
f (t ) e
r t
dt
9
Example
Let’s continue the previous example where
f (t) = 600 e0.06 t
Find the future value in 2 years at a rate of 10%.
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Example
Let’s continue the previous example where
f (t) = 600 e0.06 t
Find the future value in 2 years at a rate of 10%.
r = 0.10, T = 2, f (t) = 600 e 0.06t
FV  e
e
rT
0.10 ( 2 )
 600e

0.2

2
0

T
0
f (t ) e r t dt
600e 0.06 t e  0.10 t dt
2
0
e 0.04 t dt
 (600)(1.22140)(1.92209) 1,408.59
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Consumers’ Surplus
If ( x , p) is a point on the graph of the price-demand equation
P = D(x), the consumers’ surplus CS at a price level of p is
x
CS   [ D ( x)  p ]dx
0
which is the area between p = p
and p = D(x) from x = 0 to x = x
The consumers’ surplus represents
the total savings to consumers who
are willing to pay more than p for p
the product but are still able to buy
the product for p .
Barnett/Ziegler/Byleen Business Calculus 11e
CS
x
12
Example
Find the consumers’ surplus at a price level of p  120
for the price-demand equation
p = D (x) = 200 – 0.02x
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Example
Find the consumers’ surplus at a price level of p  120
for the price-demand equation
p = D (x) = 200 – 0.02x
Step 1. Find the demand when the price is p  120
p  200  0.02x
120  200  0.02x
x
x  4,000
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Example
(continued)
Step 2. Find the consumers’ surplus:
CS 


 D ( x)  p dx
x
0
4000
0

4000
0
(200  0.02x  120) dx
(80  0.02x) dx
 80x – 0.01x
2 4000
|
0
 320,000 – 160,000  $160,000
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Producers’ Surplus
If ( x , p) is a point on the graph of the price-supply equation
p = S(x), then the producers’ surplus PS at a price level of p is
x
p  PS   [ p  S ( x)]dx
p
0
CS
p
x
which is the area between
p  p and p = S(x) from
x = 0 to x  x
p
The producers’ surplus represents the total gain to producers
who are willing to supply units at a lower price than p but are
able to sell them at price p .
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Example
Find the producers’ surplus at a price level of p  $55
for the price-supply equation
p = S(x) = 15 + 0.1x + 0.003 2
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Example
Find the producers’ surplus at a price level of p  $55
for the price-supply equation
p = S(x) = 15 + 0.1x + 0.003x2
Step 1. Find x , the supply when the price is p  $55
p  15  0.1x  0.003x
2
55  15  0.1x  0.003x
2
2
0  0.003x  0.1x  40
Solving for x using a graphing utility: x  100
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Example
(continued)
Step 2. Find the producers’ surplus:
x  100
PS 

100

0
[55  (15  0.1 x  0.003x 2 ) ] dx
0

  p  S ( x) dx
x
100
0
(40  0.1 x  0.003x 2 ) dx
 40x – 0.05x – 0.001x
2
3
|
100
0
 4 ,000 – 500 – 1,000  $2 ,500
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Summary
■ We learned how to use a probability density function.
■ We defined and used a continuous income stream.
■ We found the future value of a continuous income stream.
■ We defined and calculated a consumer’s surplus.
■ We defined and calculated a producer’s surplus.
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