Transcript Document

4.1 Mathematical Expectation
• Example: Repair costs for a particular machine
are represented by the following probability
distribution:
x
$50
200
350
P(X = x)
0.3
0.2
0.5
• What is the expected value of the repairs?
– That is, over time what do we expect repairs to cost on
average?
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Expected value
• μ = E(X)
– μ = mean of the probability distribution
• For discrete variables,
μ = E(X) = ∑ x f(x)
• So, for our example,
E(X) = ________________________
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Your turn …
• By investing in a particular stock, a person can
take a profit in a given year of $4000 with a
probability of 0.3 or take a loss of $1000 with a
probability of 0.7. What is the investor’s
expected gain on the stock? (problem 7, pg. 94)
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Expected value of Continuous Variables
• For continuous variables,
μ = E(X) = _______
• Example: Recall from last time, problem 7 (pg. 73)
f(x) =
{
x,
2-x,
0,
0<x<1
1≤x<2
elsewhere
(in hundreds of hours.)
What is the expected value of X?
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• E(X) = ∫ x f(x) dx
= ________________________
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Functions of Random Variables
• Example 4.4. Probability of X, the number of cars passing
through a car wash in one hour on a sunny Friday afternoon,
is given by
x
P(X = x)
4
5
1/12 1/12
6
7
8
9
1/4
1/4
1/6
1/6
Let g(X) = 2X -1 represent the amount of money paid to the
attendant by the manager. What can the attendant expect to
earn during this hour on any given sunny Friday afternoon?
E[g(X)] = Σ g(x) f(x) = ____________________
= _______________________________
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4.2 Variability
• Recall our example: Repair costs for a particular
machine are represented by the following
probability distribution:
x
$50
200
350
P(X = x)
0.3
0.2
0.5
• What is the variance in the repair cost?
– That is, how might we define the spread of costs?
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Variance
• Remember
 μ = E(X) = $230
• Then, for discrete variables,
σ2 = E [(X - μ)2] = ∑ (x - μ)2 f(x)
= E (X2) - μ2
So, for our example,
σ2 = ________________________
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Your turn …
• By investing in a particular stock, a person can
take a profit in a given year of $4000 with a
probability of 0.3 or take a loss of $1000 with a
probability of 0.7. What are the variance and
standard deviation of the investor’s gain on the
stock? (problem 7, pg. 94)
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Variance of Continuous Variables
• For continuous variables,
σ2 = E [(X - μ)2] =_____________
• Example: Recall from last time, problem 7 (pg.
73)
x,
0<x<1
f(x) =
2-x,
1≤x<2
0,
elsewhere
{
(in hundreds of hours.)
What is the variance of X?
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• σ2(X) = [∫ x2 f(x) dx] – μ2
= ________________________
• What is the standard deviation?
σ = _______________
• What does this mean?
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What if the distribution is unknown?
• Chebyshev’s theorem:
The probability that any random variable X will
assume a value within k standard deviations of
the mean is at least 1 – 1/k2. That is,
P(μ – kσ < X < μ + kσ) ≥ 1 – 1/k2
• Look back at today’s examples.
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Functions of Random Variables
• Example 4.4. Probability of X, the number of cars passing
through a car wash in one hour on a sunny Friday afternoon,
is given by
x
P(X = x)
4
5
1/12 1/12
6
7
8
9
1/4
1/4
1/6
1/6
If g(X) = 2X -1 represents the amount of money paid to the
attendant by the manager, what is the variance of the amount
paid on a Friday afternoon?
σg(X)2 = E{[g(X) – μg(X)]2} = ____________________
= _______________________________
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Covariance
• A measure of the nature of the association
between two variables.
• Example 4.14:
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