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Supplemental Text Material to Support
th
Introduction to Statistical Quality Control 4 Edition
Douglas C. Montgomery
John Wiley & Sons, New York, 2001
1. Independent Random Variables
Preliminary Remarks
Readers encounter random variables throughout the textbook. An informal definition of
randomvariable
and notation for random variables is used. A may be thought of
informally as any variable for which the measured or observed value depends on a
random or chance mechanism. That is, the value of a random variable cannot be known
in advance of actual observation of the phenomena. Formally, of course, a random
variable is a function that assigns a real number to each outcome in the sample space of
the observed phenomena. Furthermore, it is customary to distinguish between the random
variable and its observed value or realization by using an upper-case letter to denote the
X
random variable (say ) and a corresponding lower-case letter for the actual numerical
value x that is the result of an observation or a measured value. This formal notation is
not used in the book because (1) it is not widely employed in the statistical quality control
field and (2) it is usually quite clear from the context whether we are discussing the
random variable or its realization.
Independent Random variables
In the textbook, we make frequent use of the concept of independent random variables.
Most readers have been exposed to this in a basic statistics course, but here a brief review
of the concept is given. For convenience, we consider only the case of continuous
random variables. For the case of discrete random variables, refer to Montgomery and
Runger (1999).
Often there will be two or more random variables that jointly define some physical
phenomena of interest. For example, suppose we consider injection-molded components
used to assemble a connector for an automotive application. To adequately describe the
connector, we might need to study both the hole interior diameter and the wall thickness
x x
of the component. Let 1 represent the hole interior diameter and 2 represent the wall
joint probability distribution
thickness. The (or density function) of these two
continuous random variables can be specified by providing a method for calculating the
x x R
probability that and assume a value in any region of two-dimensional space, where
1 2
R range space
the region is often called the of the random variable. This is analogous to
the probability density function for a single random variable. Let this joint probability
fxx
density function be denoted by (,). Now the double integral of this joint probability
12
R x x
density function over a specified region provides the probability that and assume
1 2
R
values in the range space .
A joint probability density function has the following properties:
fxxt xx
a. ( , ) 0 for all ,
12 12
1
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b. ff
fxxdxdx
(,) 1
³³ 1212
f f
R Pxx R fxxdxdx
c. For any region of two-dimensional space {(,) } (,)
12 ³³ 1212
R
x x independent fxx fxfx
The two random variables and are if (,) ()()where
1 2
12 1122
fx fx marginal x x
( ) and ( ) are the probability distributions of and , respectively,
11 22 1 2
defined as
ff
fx fxxdx fx fxxdx
() (,) and () (,)
11 122 22 121
³³
f f
p xx x
In general, if there are random variables , ,..., p then the joint probability density
12
fxx x
function is (,,..., ), with the properties:
12 p
fxx x xx x
a. ( , ,..., ) t 0 for all , ,...,
pp
12 12
f x x x dxdx dx
b. ... ( , ..., ) ... 1
pp
³³ ³ 12 12
R
R p
c. For any region of -dimensional space,
Pxx x R fxx xdxdxdx
{( , ,..., ) } ... ( , ,..., ) ...
ppp
12 ³³ ³ 12 12
R
x x x independent
The random variables , , …, are if
1 2 p
fxx x fxfx fx
(,,..., ) ()()...()
ppp
12 1122
fx x x x
where ( )are the marginal probability distributions of , , …, , respectively,
ii 1 2 p
defined as
f x f x x x dxdx dx dx dx
( ) ... ( , ,..., ) ... ...
ii p i i p
³³ ³ 12 12 11
Rx
i
2. Random Samples
To properly apply many statistical techniques, the sample drawn from the population of
randomsample x
interest must be a . To properly define a random sample, let be a
random variable that represents the results of selecting one observation from the
fx x n
population of interest. Let ( )be the probability distribution of . Now suppose that
observations (a sample) are obtained independently from the population under
unchanging conditions. That is, we do not let the outcome from one observation influence
x
the outcome from another observation. Let be the random variable that represents the
i
i xx x
observation obtained on the th trial. Then the observations , ,..., n are a random
12
sample.
2
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In a random sample the marginal probability distributions f (x ), f (x ),..., f (x )are all
12n
identical, the observations in the sample are independent, and by definition, the joint
probability distribution of the random sample is (,,...,) ()()...().
f xx x f x f x f x
nn
12 1 2
3. Development of the Poisson Distribution
The Poisson distribution is widely used in statistical quality control and improvement,
count data
frequently as the underlying probability model for . As noted in Section 2-2.3
of the text, the Poisson distribution can be derived as a limiting form of the binomial
distribution, and it can also be developed from a probability argument based on the birth
and death process. We now give a summary of both developments.
The Poisson Distribution as a Limiting Form of the Binomial Distribution
Consider the binomial distribution
n
§·
xnx
() (1 )
px p p
¨¸
x
©¹
n!
xnx
(1 ) , 0,1,2,...,
!( )! ppxn
xnx
Let O npso that p O /n. We may now write the binomial distribution as
xnx
OO
( 1)( 2) ( 1)
nn n n x n
§·§ ·
()
px ! ¨¸¨ ¸
xnn
©¹© ¹
x xn
OOªº O
12x1
§·§·§ ·§·§·
(1) 1 1 1 1 1
¨¸¨¸¨ ¸¨¸¨¸
! «»
xnnnnn
©¹©¹© ¹©¹©¹
¬¼
of o O
Let and 0so that remains constant. The terms
np np
12x1 Ox
§·§·§ ·§·
1,1,...,1 and 1 all approach unity. Furthermore,
¨¸¨¸¨ ¸¨¸
nnn n
©¹©¹© ¹©¹
O n
§·O
o of
1 as
en
¨¸
n
©¹
Thus, upon substitution we see that the limiting form of the binomial distribution is
OxeO
()
px x!
which is the Poisson distribution.
Development of the Poisson Distribution from the Poisson Process
Consider a collection of time-oriented events, arbitrarily called “arrivals” or “births”. Let
x be the number of these “arrivals” or “births” that occur in the interval [0,t). Note that
t
3
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x R
the range space of is = {0,1,…}. Assume that the number of births during non-
t
overlapping time intervals are independent random variables, and that there is a positive
constant O such that for any small time interval 't , the following statements are true:
1. The probability that exactly one birth will occur in an interval of length 't is
O't.
2. The probability that zero births will occur in the interval is .
1O 't
3. The probability that more than one birth will occur in the interval is zero.
The parameter O is often called the mean arrival rate or the mean birth rate. This type of
process, in which the probability of observing exactly one event in a small interval of
time is constant (or the probability of occurrence of event is directly proportional to the
length of the time interval), and the occurrence of events in non-overlapping time
Poisson process
intervals is independent is called a .
In the following, let
Px x px ptx
{}()(),0,1,2,...
tx
Suppose that there have been no births up to time t. The probability that there are no
t
births at the end of time +'t is
pt't O'tpt
()(1)()
00
Note that
pt'tpt
()()
00
Opt
't 0()
so consequently
pt'tpt
()()
ªº
00
c
pt
lim 0( )
'o«»
t 0 't
¬¼
Opt
0()
x t
For > 0 births at the end of time +'t we have
pt't p tOO't 'tpt
()()(1)()
xx x
1
and
pt'tpt
()()
ªº
xx
c
pt
lim x ( )
'o«»
t 0 't
¬¼
OOptpt
() ()
xx
1
Thus we have a system of differential equations that describe the arrivals or births:
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