EXPECTED VALUE AND VARIANCE
• It’s important to compute mean (expected value) and
variance of probability distribution. For example,
– Recall from our discussion on random variables and
random numbers that if we want to generate random
numbers, it may be necessary to specify mean and
variance (along with the distribution) of the random
numbers.
– Suppose that you have to decide whether or not to
make an investment that has an uncertain return. You
may like to know whether the expected return is more
than the investment.
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EXPECTED VALUE
• The expected value is obtained as follows:
n
E  X    xi p  xi 
i 1
• E(X) is the expected value of the random variable X
• xi is the i-th possible value of the random variable X
• p(xi) is the probability that the random variable X will
assume the value xi
2
EXPECTED VALUE: EXAMPLE
Example 1: Hale’s TV productions is considering producing
a pilot for a comedy series for a major television network.
While the network may reject the pilot and the series, it
may also purchase the program for 1 or 2 years. Hale’s
payoffs (profits and losses in $1000s) and probabilities of
the events are summarized below:
Reject 1 year 2 years
x
-100
50
150
p(x)
0.2
0.3
0.5
What should the company do?
3
LAWS OF EXPECTED VALUE
• The laws of expected value are listed below:
1. E c   c
2. E cX   cE  X 
3. E ( X  Y )  E  X   E Y 
E ( X  Y )  E  X   E Y 
4. E ( XY )  E  X E Y , if X and Y are independen t
• X and Y are random variables
• c is a constant
• E(X), E(Y), and E(c) are expected values of X, Y and c
respectively.
4
LAWS OF EXPECTED VALUE: EXAMPLE
Example 2: If it turns out that each payoff value of Hale’s TV
is overestimated by $50,000, what the company should
do?
5
LAWS OF EXPECTED VALUE: EXAMPLE
Example 3: Tucson Machinery Inc. manufactures Computer
Numerical Controlled (CNC) machines. Sales for the CNC
machines are expected to be 30, 36, 42, and 33 units in
fall, winter, spring and summer respectively. What is the
expected annual sales?
6
LAWS OF EXPECTED VALUE: EXAMPLE
Example 4: Let X be a random variable with the following
probability distribution:
x
-10
0.2
p(x)

Compute E 2 X  5
2
5
20
0.3
0.5

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VARIANCE
• The variance and standard deviation are obtained as
2
follows:
n
2
Variance,  X2  E  X      xi    pxi 


i 1
Standard deviation,  X   X2
•  is the mean (expected value) of random variable X
• E[(X-)2] is the variance of random variable X, expected
value of squared deviations from the mean
• xi is the i-th possible value of random variable X
• p(xi) is the probability that random variable X will assume
the value xi
8
VARIANCE: EXAMPLE
Example 5: Let X be a random variable with the following
probability distribution:
x
p(x)
-10
0.2
5
20
0.3
0.5
Compute variance.
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SHORTCUT FORMULA FOR VARIANCE
• The shortcut formula for variance and deviation are as
follows:
Shortcut formula for varian ce,  X2  E X 2    2
Standard deviation,  X   X2
•  is the mean (expected value) of random variable X
• E(X 2) is the expected value of X 2 and is obtained as
n
follows:
2
2
    x px 
E X
i 1
i
i
• xi is the i-th possible value of random variable X
• p(xi) is the probability that random variable X will assume
the value xi
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SHORTCUT FORMULA FOR VARIANCE
EXAMPLE
Example 6: Let X be a random variable with the following
probability distribution:
x
p(x)
-10
0.2
5
20
0.3
0.5
Compute variance using the shortcut formula.
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LAWS OF VARIANCE
• The laws of expected value are listed below:
1. V c   0
2. V cX   c 2V  X 
3. V  X  c   V  X 
3. V ( X  Y )  V  X   V Y 
V ( X  Y )  V  X   V Y 
• X and Y are random variables
• c is a constant
• V(X), V(Y), and V(c) are variances of X, Y and c
respectively.
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LAWS OF VARIANCE: EXAMPLE
Example 7: Let X be a random variable with the following
probability distribution:
x
p(x)
-10
0.2
5
20
0.3
0.5
Compute V 2 X  5
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