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LESSON 8: RANDOM VARIABLES
EXPECTED VALUE AND VARIANCE
Outline
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Random variables
Probability distributions
Expected value
Variance and standard deviation of a random variable
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RANDOM VARIABLES
• Random variable:
– A variable whose numerical value is determined by the
outcome of a random experiment
• Discrete random variable
– A discrete random variable has a countable number of
possible values.
– Example
• Number of heads in an experiment with 10 coins
• If X denotes the number of heads in an experiment
with 10 coins, then X can take a value of 0, 1, 2, …,
10
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RANDOM VARIABLES
– Other examples of discrete random variable: number of
defective items in a production batch of 100, number of
customers arriving in a bank in every 15 minute,
number of calls received in an hour, etc.
• Continuous random variable
– A continuous random variable can assume an
uncountable number of values.
– Examples
• The time between two customers arriving in a bank,
the time required by a teller to serve a customer,
etc.
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DISCRETE PROBABILITY DISTRIBUTION
• Discrete probability distribution
– A table, formula, or graph that lists all possible events and
probabilities a discrete random variable can assume
– An example is shown on the next slide.
– Suppose a coin is tossed twice. The events
• HH: both head
• 1H1T: One head, one tail
• TT: both tail
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DISCRETE PROBABILITY DISTRIBUTION
0.75
Probability
0.5
0.25
0
HH
1H1T
TT
Event
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HISTOGRAM OF RANDOM NUMBERS
– Suppose that the following is a histogram of 40 random
numbers generated between 0 and 1.
Frequency
Histogram
8
7
6
5
4
3
2
1
0
0.125 0.25 0.375
0.5
0.625 0.75 0.875
Random Numbers
1
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RANDOM NUMBER GENERATION
• Most software can generate discrete and continuous
random numbers (these random numbers are more
precisely called pseudo random numbers) with a wide
variety of distributions
• Inputs specified for generation of random numbers:
– Distribution
– Average
– Variance/standard deviation
– Minimum number, mode, maximum number, etc.
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RANDOM NUMBER GENERATION
• Next 4 slides
– show histograms of random numbers generated and
corresponding input specification.
– observe that the actual distribution are similar to but
not exactly the same as the distribution desired, such
imperfections are expected
– methods/commands used to generate random
numbers will not be discussed in this course
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: uniform distribution,
min = 500 and max = 800
Frequency
Uniform Distribution
25
20
15
10
5
0
Random Numbers
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: triangular distribution,
min = 3.2, mode = 4.2, and max = 5.2
Frequency
Triangular Distribution
120
100
80
60
40
20
0
3.2 3.4 3.6 3.8
4
4.2 4.4 4.6 4.8
Radom Numbers
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5.2
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: normal distribution,
mean = 650 and standard deviation = 100
Random Numbers
0
95
0
91
0
87
0
83
0
79
0
75
0
71
0
67
0
63
0
59
0
55
0
51
0
47
0
43
0
39
0
30
25
20
15
10
5
0
35
Frequency
Normal Distribution
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RANDOM NUMBER GENERATION: EXAMPLE
– A histogram of random numbers: exponential
distribution, mean = 20
Exponential Distribution
30
20
10
Random Numbers
76
71
66
61
56
51
46
41
36
31
26
21
16
11
6
0
1
Frequency
40
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CONTINUOUS PROBABILITY DSTRIBUTION
• Continuous probability distribution
– Similar to discrete probability distribution
– Since there are uncountable number of events, all the
events cannot be specified
– Probability that a continuous random variable will assume
a particular value is zero!!
– However, the probability that the continuous random
variable will assume a value within a certain specified
range, is not necessarily zero
– A continuous probability distribution gives probability
values for a range of values that the continuous random
variable may assume
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f(x)
CONTINUOUS PROBABILITY DSTRIBUTION
z
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f(x)
CONTINUOUS PROBABILITY DSTRIBUTION
z
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CONTINUOUS PROBABILITY DSTRIBUTION
• The Probability Density Function
– The probability that a continuous random variable will fall
within an interval is equal to the area under the density
curve over that range
Pa  X  b   f x dx
b
f(x)
a
a
z
b
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CONTINUOUS PROBABILITY DSTRIBUTION
Example 1: Consider the random variable X having the
following probability density function:
x
0 x2
f x    2
0 Otherwise
Plot the probability density curve and find P0.5  X  1
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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    X i p X i 
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
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EXPECTED VALUE: EXAMPLE
Example 2: 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?
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EXPECTED VALUE: EXAMPLE
Example 2:
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LAWS OF EXPECTED VALUE
• The laws of expected value are listed below:
1. E c   c
2. E cX   cE  X 
3. E c'cX   c'cE  X 
4. E ( X  Y )  E  X   E Y 
E ( X  Y )  E  X   E Y 
5. E ( XY )  E  X E Y , if X and Y are independen t
• X and Y are random variables
• c, c’ are constants
• E(X), E(Y), and E(c) are expected values of X, Y and c
respectively.
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LAWS OF EXPECTED VALUE: EXAMPLE
Example 3: If it turns out that each payoff value of Hale’s TV
is overestimated by $50,000, what the company should
do? Use the answer of Example 2 and an appropriate law
of expected value. Which law of expected value applies?
Check.
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LAWS OF EXPECTED VALUE: EXAMPLE
Example 4: 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? Which law of expected value do
you use to answer this?
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VARIANCE
• The variance and standard deviation are obtained as
2
follows:
n
2
Variance,  X2  E  X       X i    p X i 


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
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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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VARIANCE: EXAMPLE
Example 5:
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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
•  = E(X ), 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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SHORTCUT FORMULA FOR VARIANCE
EXAMPLE
Example 6:
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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 c  X   V  X 
4. V c'cX   c 2V  X 
5. V ( X  Y )  V  X   V Y 
V ( X  Y )  V  X   V Y 
• X and Y are random variables
• c, c’ are constants
• 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
Which law of variance applies? Check.
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LAWS OF VARIANCE: EXAMPLE
Example 7:
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LAWS OF EXPECTED VALUE: EXAMPLE
Example 8: Let X be a random variable with the following
probability distribution:
x
-10
0.2
p(x)

5
20
0.3
0.5

Compute E 2 X  5
Which laws of expected value and variance apply? Check.
2
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LAWS OF EXPECTED VALUE: EXAMPLE
Example 8:
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EXPECTED VALUE AND VARIANCE
OF A CONTINUOUS VARIABLE
Expected value of a continuous variable:

E X    xf x dx

Variance of a continuous variable:

Var  X    x 2 f x dx  E X 
2

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READING AND EXERCISES
Lesson 8
Reading:
Section 7-1, 7-2, pp. 191-202
Exercises:
7-4, 7-6, 7-15
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