Download Mini-Excursion 4

16 Mathematics of Managing Risks
• Weighted Average
• Expected Value
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 1
Weighted Average or Weighted Mean
The weighted average (or weighted
mean) of a set of N numbers
v1 , v2 ,..., vN
each of which is assigned a weight
w1 , w2 ,..., wN
where w1  w2      wN  1 is:
v1  w1  v2  w2      vN  wN
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 2
Examples
If homework/quiz average is weighted
20%, 2 exams are weighted 25% each,
and final exam is weighted 30% and a
student makes homework/quiz average
87, exam scores of 80 and 92, and final
exam score 85. Compute the weighted
average.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 3
Examples
The weighted average is
0.20 (87 )  0.25(80 )  0.25(92 )  0.30 (85)  85.9
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 4
Random Variable
A random variable is a letter (X) that
denotes a single numerical value which is
observed when performing a random
experiment.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 5
Examples of Random Variable
• Toss a coin 3 times and count the number
of heads. Denote the total number of
heads by the random variable X.
• A basketball player shoots two consecutive
free throws. Denote the total number of
points scored by the random variable X.
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Excursions in Modern Mathematics, 7e: 16.5 - 6
Probability Distribution
A probability distribution for a random
variable X gives the probability for any
value of X. (Note: this is similar to a
probability assignment for a sample space)
Example:
Toss a coin 3 times and count the number
of heads. Denote the total number of
heads by the random variable X. What is
the probability distribution for X?
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 7
Probability Distribution
X
0
1
2
3
P(X)
1/8 = 0.125
3/8 = 0.375
3/8 = 0.375
1/8 = 0.125
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 8
Expected Value of a Random Variable
The expected value (E) of a random
variable X which has N possible outcomes
x1 , x2 ,..., xN
each of which is assigned a probability
p1 , p2 ,..., pN
where
p1  p2      pN  1 is:
E  x1  p1  x2  p2      xN  pN
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 9
Expected Value of a Random Variable
• The formula for the expected value is
similar to a weighted average formula.
• The expected value of a random variable X
gives the approximate value of X that
would result after repeating the random
experiment many, many times.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 10
Example
Toss a coin 3 times and count the number
of heads. Denote the total number of
heads by the random variable X. What is
the expected value of X? (Use the
probability distribution in the previous
example.)
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 11
Example
X
0
1
2
3
P(X)
1/8 = 0.125
3/8 = 0.375
3/8 = 0.375
1/8 = 0.125
E  0.125 (0)  0.375 (1)  0.375 (2)  0.125 (3)  1.5
That is, we expect there will be 1.5 heads in three tosses
(that is, we expect that 50% of the tosses would result in
heads).
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 12
Example
page 621
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Excursions in Modern Mathematics, 7e: 16.5 - 13
Example
• X is a random variable that represents the
net gain (or loss) of your bet.
• Probability distribution of X is (assuming
each guess equally likely):
X
-$1
$36
P(X)
37/38
1/38
37
1
1
E
(1)  (36 )  
 0.03
38
38
38
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 14
Example
The negative indicates that if the random
experiment were repeated many times,
there would be a net loss of about $0.03
(house wins).
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Excursions in Modern Mathematics, 7e: 16.5 - 15