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Unit 10 Expectation
Unit 10 Expectation
IT Disicipline
ITD1111 Discrete Mathematics & Statistics
STDTLP
1
Unit 10 Expectation
10.1 Mathematical expectation
In each trial of an experiment, a random
variable X may assume any value in the
range space R. In the long-run, an average
and a variance may be associated with X.
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.1-1 (1/4)
The length in meters, X, of metal bars
produced in a machine is such that
P(X < 4.5) = 0.1
P(4.5  X  5) = 0.7
P(X > 5) = 0.2.
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.1-1 (2/4)
If 4.5  X  5, a bar will be accepted by the
customers with a profit of $52.
If X < 4.5, a bar will be discarded with a
loss of $20.
If X > 5, a bar will be cut to the specified
length and the profit will be reduced to $48.
Find the expected profit.
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.1-1 (3/4)
Consider N bars be produced and
let A be the no. of bars accepted,
R be the no. of bars rejected, and
C be the no. of bars that have to be cut.
52 A  (20) R  48C
Average profit P =
N
A
R
C
= 52 N  20 N  48 N
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.1-1 (4/4)
A
R
C
P  52 lim
 20 lim
 48 lim
 Nlim

N  N
N  N
N  N
= 52(0.7)  20(0.1)  48(0.2)
= 44
The expected profit is therefore $44.
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
10.2 Expected value (definition)
In general, let X be a discrete r.v. with
possible values x1, x2, …, xn and
corresponding probabilities p1, p2, …, pn.
The expectation of X is defined as
E(X) =
 xp( x) = x1p1 + x2p2 + … + xnpn .
x
The expectation of X is also called the
expected value of X.
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.2-1
The random variable X has the distribution
shown below
x
1
2
3
4
P(X = x) 0.3 0.2 0.4 0.1
The expected value of X, E(X)
= (1)(0.3) + (2)(0.2) + (3)(0.4) + (4)(0.1)
= 2.3
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.2-2 (1/2)
The random variable X has a probability
distribution given by
c
P(X = x) =
, x = 1, 2, 3, 4, 5.
x
(a) Find c.
(b) Find E(X)
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.2-2 (2/2)
1 1 1 1

(a) c1       1
2 3 4 5

60
c=
137
c
c
c
c
(b) E(X) = (c)(1)  ( )( 2)  ( )(3)  ( )( 4)  ( )(5)
2
3
4
5
= 5c
300
=
137
IT Disicipline
ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.2-3
In a game, a player tosses 3 fair coins. He
wins $10 if 3 heads occur, $5 if 2 heads
occur, $2 if only 1 head occurs and losses
$15 if no heads occur. What is his expected
gain?
His expected gain
= 10(1/8)+5(3/8)+3(3/8)-15(1/8)
= 2 (dollars)
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
10.3 Function of a random
variable
If X is a discrete random variable with
probability distribution p(x) and if g(x) is a
real-valued function of X, then the expected
value of g(X) is defined as
E[g(X)] =
 g ( x) p( x)
x
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.3-1 (1/3)
A salesman is employed by a computer
manufacturer to sell PC’s. The salary he
gets in a day is calculated by the formula
g(x) = 90 + 60x
where x is the number of PC’s he sells in
that day.
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.3-1 (2/3)
Assume that in each day he may sell zero to
four PC’s with probabilities listed in the
table below:
Number of PC’s, x 0
1 2 3
4
Probability, p(x) 0.05 0.2 0.4 0.2 0.15
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.3-1 (3/3)
Let X be the number of PC’s sold in a day.
His expected daily salary is
E[g(X)] = g(0)p(0) + g(1)p(1) + g(2)p(2) +
g(3)p(3) +g(4)p(4)
= (90)(0.05) + (90+60)(0.2) +
(90+120)(0.4) + (90+180)(0.2) +
(90+240)(0.15)
= 222 (dollars)
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Expectation of aX + b
It can be proved that, for any real numbers
a and b,
E(aX + b) = aE(X) + b .
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.3-2
In Example 7.3-5,
expected daily salary
= E(90+60X)
= 90 + 60E(X)
= 90 + 60[(0)(0.05) + (1)(0.2) + (2)(0.4)
+ (3)(0.2) + (4)(0.15)]
= 222 (dollars)
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
10. 4 Variance of a random
variable
The variance of a random variable X is
defined as
V(X) = 2 = E[(X –)2] ,
where  = E(X).
The standard deviation, , of a random
variable X is defined
=
IT Disicipline
E[( X   ) ] .
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.4-1 (1/2)
The probability distribution of the random
variable X is shown below:
X
P(X = x)
1
2
3
0.1 0.5 0.4
Find E(X) and V(X).
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ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.4-1 (2/2)
xp( x)
E(X) = 
x
= 1(0.1) + 2(0.5) + 3(0.4)
= 2.3
V(X) = E[(X –)2]
  ( x   ) p ( x)
x
= (1 – 2.3)2(0.1) + (2 – 2.3)2(0.5) + (3 – 2.3)2(0.4)
= 0.41
IT Disicipline
ITD1111 Discrete Mathematics & Statistics
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Unit 10 Expectation
Example 10.4-2 (1/2)
Daily sales records for a shop selling electric
appliances show that it will sell zero, one, two or
three air-conditioners with the probabilities:
Number of Sales 0
1
2
3
Probability 0.5 0.3 0.15 0.05
Calculate the expected value, variance and
standard variation for daily sales.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics
STDTLP
21
Unit 10 Expectation
Example 10.4-2 (2/2)
Expected value
= (0)(0.5) + (1)(0.3) + (2)(0.15) + (3)(0.05)
= 0.75
Variance
= (0 - 0.75)2(0.5) + (1 – 0.75)2(0.3) +
(2 – 0.75)2(0.15) + (3 – 0.75)2(0.05)
= 0.7875
Standard deviation = 0.7875 = 0.8874
IT Disicipline
ITD1111 Discrete Mathematics & Statistics
STDTLP
22