IE 331
Chapter 4
Mathematical Expectation
Dr. Waleed Mirdad
Part 1
Sample
X
x
x
x
1
I
population
mean
Sample mean
I
Sample variance
S2
M
mass
t.ma.im
or
Population
aen.igema.im
Sam le
y
i
x
x
x
X
for
length
Saudi
density function
i
4
sax
2
in
-
I
m
f
s
on
stnoh
Mean of a Random Variable
The mean of the random variable X or the mean of the probability
distribution of X can be written as μx or simply as μ when it is clear to
which random variable we refer.
17
Population mean
The Expected Value (The mean) of a Discrete
Random Variable
Definition 4.1 [Page 112]
Let X be a discrete random variable with probability distribution f(x).
The mean, or expected value, of X is
S
X
fex
M
Population
mean
True mean
Expected
value
E Cx
X
X
Xz
fck fox
fCX
Xz f Xz
o
1
TT
p
Example
PCHT
L
2 Lz
or P TH
on
f
T
4
Etta Exit
Ina
Assume that one fair coin was tossed twice.
TT
• a. List the sample space.
• b. Let X be the number of head appears. Find the probability distribution of X.
• c. Find the mean of X.
FEET
X
This
result
avera
off
ly
TH
I
l
f Cx
M
HT
TT
Sample point
E
X
means that
9
head
O
f
2
k
I
HH
iz
E E
True mean
ly
L
2
t
a person who tosses
2 Coins over and over again will 04
toss
4
H
T
HT
I
TH
I
as
S
H
S
s
g
X
H
H
H
Z
T
s
T
TT
g
Ly
PCHH
L
t
t
4
Example
A lot containing 7 components is sampled by a quality inspector; the lot
contains 4 good components and 3 defective components. A sample of
3 is taken by the inspector. Find the expected value of the number of
C Cx
good components in this sample.
M
X represents the
in the
sample
S
X
fcN
number of
good component
DDD
O
NNN
I
2
7Toj
1 Sample space
3
96
a
D
D
46
D
3
N
n
7
N
D
Ey
y
4g
D
I
DD N
D.D
36 210
ND D
I
71
t
II
24 Caio
N
2
3612210
36 210
N ND
3 N N
2,10
1210
Z D NN
2
0
24
24 210
2 N D
Ec
I Is
1 D ND
I
316
O
24
N
210,38
210
3
21
20
PM F
Example
A salesperson for a medical device company has two appointments on a given day.
At the first appointment, he believes that he has a 70% chance to make the deal,
a
from which he can earn $1,000 commission if successful.
On the other hand, he
thinks he only has a 40% chance to make the deal at the second appointment, from
which, if successful, he can make $1,500. What is his expected commission based
on his own probability belief? Assume that the appointment results are
D
independent of each other.
Deal
Li
Lose
X
SpgffY
L
L
O
X
FCK
How
s
CZ
much
the salesperson get
D
LD
1500
12
1000
42
1000
1500
28
70
p C DD
DD
D
28
40
z soo
mo
D
L
LD
D
j
L
30
L
60
42
30 1.40
12
70
18 X
O
x
c
l SOO
28
2500
t
12
1500
t
Z SOO
O
t
1
o
SO
c
42
1300
1300 Mean 2
2ndday 3rd
day
2500
O
oooo
1Stday
O
1500
18
LL
y E
what does
100 .70
P DL
DL
Go
soo
o o ee cud
a
days
a
1300
I
300
So
if
we
repeat
probability
40
Success
Long
time
70
for
let's
the sales will earn
the
same appointments with
Success for
2nd
say
1300
the est
appointment
300
Per
times
day
same
appointment
and
for
run
then
Long
on
average
The Expected Value of a Discrete Random Variable
Theorem 4.1 [Page 114]
Let X be a random variable with probability distribution f(x). The mean,
or expected value, of X is
Let X be a random variable with probability distribution f(x). The
expected value of the random variable g(x) is
Example
Suppose that the number of cars that pass through a car wash between 4:00 P.M.
and 5:00 P.M. on any sunny Friday, X, has the following probability distribution:
K
I
9
7
13
2
as
a
Let g(x) = 2X-1 represent the amount of money, in dollars, paid to the attendant by
the manager. Find the attendant’s expected earnings for this particular time period.
C Cx
is
7
Iz
12
67
t
9 t
Iz
t
Zz
t
4K
Means of Linear Combinations of Random
Variables
If a and b are constants, then E(aX + b) = aE(X) + b.
Example
Resolve the previous example example by applying the means of linear
combinations of Random Variables to the discrete random variable g(x)
= 2X-1.
E
2X
C CX
C
4
2x
2
l
1
42
5
C CX
l
12
2C 6 83
94,22
I
12.67
6 83
The Expected Value (the Mean) of a Continuous
Random Variable
Definition 4.1 [Page 112]
Let X be a continuous random variable with probability distribution f(x).
The mean, or expected value, of X is
h
I
Example
Let X be the random variable that denotes the life, in hours, of a certain
electronic device. The probability density function is:
PDF
Example
too
foxy
e
E Cx
where
3
2010002
x
113
100
DX
an
29000J
IS
1
co
20,000
DX
J
IZ
DX
100
100
co
0
I
20
Joo
20,000
000
I
160
000
20
OO
200
The Expected Value (the Mean) of a Continuous
Random Variable
Definition 4.1 [Page 112]
Let X be a continuous random variable with probability distribution f(x).
The mean, or expected value, of X is
Let X be a random variable with probability distribution f(x). The mean,
or expected value, of X is
Example
The random variable X (the error in the reaction temperature, in ◦C, for
a controlled laboratory experiment) has the following probability
density function:
Find the expected value of g(x) = 4X + 3.
If
4
3
731
DX
8
Integration
details
The Linear Property of Expected Value
Means of Linear Combinations of Random Variables
If a and b are constants, then E(aX + b) = aE(X) + b.
Example (Continuous Random Variable)
Resolve the previous example by applying the property of linear
combination to the continuous random variable g(x) = 4X+3.
E C TX
E
Cx
E C 4
4
3
132
ECD
X
FI
4
I
c 3
DX
25
I
3
25
8
Integration
details
The Linear Property of Expected Value
The expected value of the sum or difference of two or more functions
of a random variable X is the sum or difference of the expected values
of the functions. That is
Example
E C 43
Ec x3
XZ
Ect
3
c
2
X
ZECH
l X
o
EG
EC
I'M
of
Example
f
f
ga
DX
The weekly demand for a certain drink, in thousands of liters, at a chain
of convenience stores is a continuous random variable g(x) = X2 +X -2,
where X has the density function:
Find the expected value of the weekly demand for the drink.
E
CH
X
ECK
2
E CX2
E
E C
112
CX
t
Z
if
2
2
83
ECD
2C X
s
Cx
1
I
6
DX
ECD
2
83
67
DX
2
Integration
detail
es e
Variance and of Discrete Random Variables
Let X be a random variable with probability distribution f(x) and mean
μ. The variance of X is
Or
The positive square root of the variance, σ, is called the standard
deviation of X.
Example
Tfcx
Let the random variable X represent the number of defective parts for a
machine when 3 parts are sampled from a production line and tested.
The following is the probability distribution of X:
12
0
9
4
I
E
calculate σ2.
E
X
Me
Ect
ok
O
S1
x
Sl
ECE
1
38
11 1.38
Ect
ooo
13
01
914.01
44.10
87
61
61
2
187
4979
Variance and of Continuous Random Variables
Let X be a continuous random variable with probability distribution f(x)
and mean μ. The variance of X is
Or
The positive square root of the variance, σ, is called the standard
deviation of X.
Example
d
Ec
The weekly demand for a drinking-water product, in thousands of
liters, from a local chain of efficiency stores is a continuous random
variable X having the probability density:
Find the mean and variance of X.
ECD
f
ECE
if
o2
X
112
Z
a
X
2 Cx
I
DX
l
2
1
DX
The Linear Property of Variance
• The variance of Linear function of X:
Variance(aX+b) =a2 Variance(X)
var
Ca
x
16
a
2
van
C X
Need
Example