Transcript 14.2
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) =
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c
f ( x) dx
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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.
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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 )
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dt e
rT
T
0
f (t ) e
r t
dt
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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%.
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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 .
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CS
x
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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
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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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