Lec5 computer simulation Modeling and Simulation.pdf

Review of
Basic Probability and Statistics

4.1 Introduction
Probability
and
statistics
Model a
probabilistic
system
Validate the
simulation
model
Choose the
input probabilistic
distribution
Generate random
samples from the
input distribution
Perform statistic
analyses of the
simulation output data
Design the
simulation
experiments
2

4.2 Random variables and their
properties
• Experiment is a process whose outcome is not
known with certainty.
• Sample space (S) is the set of all possible
outcome of an experiment.
• Sample points are the outcomes themselves.
• Random variable (X, Y, Z) is a function that
assigns a real number to each point in the
sample space S.
• Values x, y, z
3

Examples
• Experiment consist of flipping a coin, then
S={H, T}, where {} means the “set consisting of” and “H” and “T” means
the outcome is a head and a tail, respectively.
• Experiment consist of tossing a die, then
S={1,2,…,6}, where the outcome I means that I appeared on the die. I = 1,
2, 3, …6.
A random variable is a function (or rule) that assigns a real number (any
number greater than - ∞ and less than ∞) to each point in the sample space S.
• Experiment consist of flipping two coins, then
S={(H,H), (H,T), (T,H), (T,T)}
X: the number of heads that occurs
• Experiment consist of rolling a pair of dice.
S={(1,1), (1,2), …, (6,6)}, where (I, j) means that I and j appeared on the
first and second die, respectively. If X is a random variable corresponding
to the sum of the two dice, then X assigns the value 7 to the outcome (4,3).
4

Distribution (cumulative) function
F(x) of the random variable X is defined for each real
number x as:
)
( x
X
P 






 x
x
X
P
x
F for
)
(
)
(
}
{ x
X 
: the probability associated with the event
Properties:
1.
2. F(x) is nondecreasing [i.e., if ].
3. .
.
all
for
1
)
(
0 x
x
F 

)
(
)
(
then
, 2
1
2
1 x
F
x
F
x
x 

0
)
(
lim
and
1
)
(
lim
x






x
F
x
F
x
5

Discrete random variable
A random variable X is said to be discrete if it can take on at most a
countable number of values, say x1, x2, x3, ….
The probability that the discrete random variable X takes on the value is
given by
Then
Probability mass function (p(x) for discrete random variable X ) on I=[a,b],
a and b are real numbers such as a ≤ b, then
i
x
,...
2
,
1
for
)
(
)
( 

 i
x
X
P
x
p i
i




1
1
)
(
i
i
x
p













x
x
p
x
F
x
p
I
X
P
x
x
i
b
x
a
i
i
i
all
for
)
(
)
(
)
(
)
(
6

Examples: for the inventory example, the size of the demand for
the product is a discrete random variable X that takes on the
values 1, 2, 3, 4 with respective probabilities 1/6, 1/3, 1/3, 1/6.
The probability mass function and the distribution function for X
are given below:
0 1 2 3 4 x
p(x)
1
1/6
1/3
1/2
2/3
5/6
p(x) for the demand-size random variable X.
3
2
3
1
3
1
)
3
(
)
2
(
)
3
2
( 





 p
p
X
P
7

0 1 2 3 4 x
F(x)
1
1/6
1/3
1/2
2/3
5/6
F(x) for the demand-size random variable X.
8

Continuous random variables
dx
x
f
B
X
P B
)
(
)
( 


A random variable is said to be continuous if there exists a nonnegative function f(x)
such that for any set of real number B
and 1
)
( 




dx
x
f
F(x) is the probability distribution function for the continuous RV X is given by
0
)
(
])
,
[
(
)
( 




 dy
y
f
x
x
X
P
x
X
P
x
x
dy
y
f
x
x
x
X
P
x
x
x
)
(
])
,
[
( 






dy
y
f
x
X
P
x
F
x
)
(
])
,
(
(
)
( 



 

for all 



 x
)
(
)
( x
F
x
f 

)
(
)
(
)
(
)
( a
F
b
F
dy
y
f
I
X
P
b
a




 ]
,
[ b
a
I 
f(x) is called the probability density function for continuous random variable X.
If I = [a, b] for any real numbers a and b such that a < b, then
9

x
f(x)
x
x 
 x
x 

'
'
x
x
])
,
[
( x
x
x
X
P 


])
,
[
( '
'
x
x
x
X
P 


Interpretation of the probability density function
10





b
a
dy
y
f
b
x
a
P )
(
)
(
a b
x
f(x)
Distribution and density functions of a Continuous Random Variable
Given an interval I = [a,b]
)
(
)
(
)
(
)
( a
F
b
F
dy
y
f
I
X
P
b
a



 
11

Example: Uniform random variable on the interval [0,1]
otherwise
0
1
0
if
1
{
)
(



x
x
f
1
0 
 x
If
x
dy
dy
y
f
x
F
x
x




 1
)
(
)
( 0
0
, then
12

0 x
f(x)
1
1
f(x) for a uniform random variable on [0,1]
otherwise
0
1
0
if
1
{
)
(



x
x
f
0 x
F(x)
1
1
F(x) for a uniform random variable on [0,1]
x
x
x
x
x
F
x
x
F
dy
y
f
x
x
x
X
P
x
x
x

















)
(
)
(
)
(
)
(
])
,
[
(
1
0 



 x
x
x
where
13

Example: Exponential random variable
0 x
f(x)

1
0 x
F(x)
1
f(x) for an exponential random
variable with mean 
F(x) for an exponential random
variable with mean 
14

Joint Random Variables
In the M/M/1 queuing system, the input can be represented as two sets of
random variables:
arrival times of customers: A1, A2, …, An
and service times of customers: S1, S2, …, Sn
The output can be a set of random variables:
delays in queue of customers: D1, D2, …, Dn
The D’s are not independent.
15

Jointly discrete Random Variables
Joint probability mass function
If X and Y are discrete random variables, then let
)
,
(
)
,
( y
Y
x
X
P
y
x
p 

 for all x, y
where p(x,y) is called the joint probability mass function of X and Y.
X and Y are independent if
)
(
)
(
)
,
( y
p
x
p
y
x
p Y
X
 for all x, y
where
)
,
(
)
(
all
y
x
p
x
p
y
X 

)
,
(
)
(
all
y
x
p
y
p
x
Y 

are the (marginal) probability mass functions of X and Y.
16

Example 4.9
Suppose that X and Y are jointly discrete random variables with
otherwise
0
4
,
3
,
2
and
2
,
1
for
{
)
,
( 27 


y
x
y
x
p
xy
Then
3
27
4
2
)
( x
xy
y
X x
p 



9
27
2
1
)
( y
xy
x
Y y
p 



for x=1,2
for y=2,3,4
Since )
(
)
(
27
/
)
,
( y
p
x
p
xy
y
x
p Y
X

 For all x, y, the random variables X and Y
are independent.
17

• Suppose that 2 cards are dealt from a deck of 52 without
replacement. Let the random variables X and Y be the number of
aces and kings that occur, both of which have possible values of 0, 1,
2. It can be shown that
pX(1) = pY(1) = 2
And
p(1,1) = 2
Since
p1,1) = 2 ≠ 4
It follows that X and Y are not independent.
Example 4.10
)
)(
( 51
48
52
4
)
)(
( 51
4
52
4
)
)(
( 51
4
52
4 2
51
48
2
52
4
)
(
)
(
18

Jointly continuous Random Variables
Joint probability density function
The random variables X and Y are jointly continuous if there exists a nonnegative
function f(x,y), such that for all sets of real numbers A and B,
dxdy
y
x
f
B
Y
A
X
P A
B
)
,
(
)
,
( 




X and Y are independent if
)
(
)
(
)
,
( y
f
x
f
y
x
f Y
X
 for all x and y
where
dy
y
x
f
x
fX )
,
(
)
( 




dx
y
x
f
y
fY )
,
(
)
( 




are the (marginal) probability density functions of X and Y, respectively.
19

Example 4.11
Suppose that X and Y are jointly continuous random variables with
otherwise
0
1
and
,
0
,
0
for
24
{
)
,
(





y
x
y
x
xy
y
x
f
Then
2
1
0
2
1
0
)
1
(
12
12
24
)
( x
x
xy
xydy
x
f x
x
X 



 

for 1
0 
 x
2
1
0
2
1
0
)
1
(
12
12
24
)
( y
y
yx
xydx
y
f y
y
Y 



 

1
0 
 y
for
Since
)
(
)
(
)
(
6
)
,
( 2
1
2
1
2
2
3
2
1
2
1
Y
X f
f
f 


X and Y are not independent.
20

Mean or expected value














continuous
is
if
)
(
discrete
is
if
)
(
)
( 1
i
X
i
j
X
j
j
i
i
X
x
xf
X
x
p
x
x
E
i
i

The mean is one measure of central tendency in the sense that
it is the center of gravity
21

Examples 4.12-4.13
For the demand-size random variable,(the size of the demand for the product is
a discrete random variable X that takes on the values 1, 2, 3, 4 with respective
probabilities 1/6, 1/3, 1/3, 1/6) the mean is given by
2
5
)
6
1
(
4
)
3
1
(
3
)
3
1
(
2
)
6
1
(
1 





For the uniform random variable on the interval [0, 1] has the following pdf
2
1
)
(
1
0
1
0




 xdx
dx
x
xf

otherwise
0
1
0
if
1
{
)
(



x
x
f
1
0 
 x
If
x
dy
dy
y
f
x
F
x
x




 1
)
(
)
( 0
0
, then
The mean is given by
22

Properties of means
)
(
)
( X
cE
cX
E 
)
(
)
( 1
1 i
i
n
i
i
i
n
i X
E
c
X
c
E 

 

Xi
1.
Even if the ‘s are dependent.
2.
Let c or ci denote a constant (real number), then the
following are important properties of means:
23

Median
The median of the random variable is defined to be the
smallest value of x such that FX(x) ≥ 0.5. If Xi is a
continuous random variable, then
x0.5
x0.5
5
.
0
)
( 5
.
0 
x
F i
X
x0.5 x
)
(x
f i
X
area=0.5
The median for a continuous random variable
x0.5
The median may be better measure of central tendency than mean when Xi can
Take on very large or very small values.
24

Example 4.14
The median may be a better measure of central tendency than the mean.
1. Consider a discrete random variable X that takes on each of the
values, 1, 2, 3, 4, and 5 with probability 0.2. Clearly, the mean
And the median of X are 3.
2. Now consider random variable Y that takes on each of the
values, 1, 2, 3, 4, and 100 with probability 0.2. The mean
and the median of X are 22 and 3, respectively.
Note that the median is insensitive to this change in the distribution.
25

Variance
]
)
[(
)
( 2
2
i
i
i
i X
E
X
Var 
 

 2
2
2
2
)]
(
[
)
(
)
( i
i
i
i X
E
X
E
X
E 


 
For the demand-size random variable,
6
43
)
6
1
(
4
)
3
1
(
3
)
3
1
(
2
)
6
1
(
1
)
( 2
2
2
2
2





X
E
12
11
)
2
5
(
6
43
)
(
)
( 2
2
2




 
X
E
X
Var
For the uniform random variable on [0,1],
3
1
)
(
)
( 2
1
0
2
1
0
2




 dx
x
dx
x
f
x
X
E
12
1
)
2
1
(
3
1
)
(
)
( 2
2
2




 
X
E
X
Var
The variance of the random variable Xi will be denoted by
and is defined by
)
(
2
i
i X
orVar

26

 
2

large
2

small
Density functions for continuous random variables with
large and small variances.
27

Properties of the variance
0
)
( 
X
Var
)
(
)
( 2
X
Var
c
cX
Var 
)
(
)
( 1
1 i
n
i
i
n
i X
Var
X
Var 

 

1.
2.
3. Xi
if the ‘s are
independent (or uncorrelated).
28

Standard deviation
i
i 
 96
.
1

2
i
i 
 
i
X i
i 
 96
.
1

The probability that is between and is 0.95.
The standard deviation of the random variable Xi is
denoted to be.
29

Covariance
j
i
j
i
j
j
i
i
ij
j
i X
X
E
X
X
E
C
X
X
Cov 


 




 )
(
)]
)(
[(
)
,
(
The covariance between the random variables and
is a measure of their dependence.
Xi Xj
ji
ij C
C 
2
i
ji
ij C
C 

 if i=j,
The covariance between the random variables and
(where I = 1, 2, …, n; j = 1, 2, …, n) which is a measure of
their dependence, will be denoted by Cij or Cov(Xi, Xj) and is
defined by
Xi Xj
30

Example 4.17
For the jointly continuous random variables X and Y in Example 4.11
15
2
)
1
(
8
)
24
(
)
,
(
)
(
3
2
1
0
2
1
0
2
1
0
1
0
1
0












dx
x
x
dx
dy
y
x
dydx
y
x
xyf
XY
E
x
x
5
2
)
1
(
12
)
(
)
( 2
2
1
0
1
0





 dx
x
x
dx
x
xf
X
E X
5
2
)
1
(
12
)
(
)
( 2
2
1
0
1
0





 dy
y
y
dy
y
yf
Y
E Y
75
2
)
5
2
)(
5
2
(
15
2
)
(
)
(
)
(
)
,
(





 Y
E
X
E
XY
E
Y
X
Cov
31

If and are independent random variables
0

ij
C
Xi Xj
Xj
Xiand are uncorrelated.
Generally, the converse is not true.
32

Correlated
If , then and are said to be positively correlated.
Xi Xj
0

ij
C
If , then and are said to be negatively correlated.
Xi j
X
0

ij
C
i
i
X 
 j
j
X 

i
i
X 
 j
j
X 

and tend to occur together
i
i
X 
 j
j
X 

i
i
X 
 j
j
X 

33

Correlation
2
2
i
i
ij
ij
C


 
n
j
n
i
,
,
2
,
1
,
,
2
,
1




1
1 

 ij

ij

If is close to +1, then and are highly positively correlated.
Xi Xj
ij

If is close to -1, then and are highly negatively correlated.
Xi Xj
For the random variable in Example 4.11
25
1
)
(
)
( 
 Y
Var
X
Var
3
2
)
(
)
(
)
,
(
)
,
(
25
1
75
2





Y
Var
X
Var
Y
X
Cov
Y
X
Cor
34

Lec5 computer simulation Modeling and Simulation.pdf

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