A GEOMETRIC STUDY OF THE EXACT SAMPLING DISTRIBUTION OF STANDARD DEVIATIONS WHEN THE SAMPLED POPULATION IS ARBITRARY

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DOCTORAL DISSERTATION SERIES Publication No,: 13,814

A GEOMETRIC STUDY OF THE EXACT SAMPLING DISTRIBUTION OF STANDARD DEVIATIONS WHEN THE SAMPLED POPULATION IS ARBITRARY

A Thesis Submitted to the Faculty of Purdue University

by Paul Eugene Iriek

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy February, 1950

P U R D U E UNIVERSITY

TH IS IS TO CERTIFY THAT THE TH ESIS PREPARED UNDER MY S U P E R V I S I O N

51__________ Paul_Eugene

Irlck

A GEOMETRIC STUDY OF THE EXACT S A M P L IN G

e n title d

D IS T R IB U T IO N OF STANDARD D E V IA T IO N S WHEN THE SAMPLED POPULATION I S A R B IT R A R Y COM PLIES WITH TH E UNIVERSITY REGULATIONS ON G R A D U A T IO N T H E S E S

AND IS APPRO V ED BY ME AS FULFILLING THIS PART O F T H E R E Q U I R E M E N T S

FO R THE D EG REE O F

D o c to r o f P h ilo s o p h y

0

$f.

■

.........

■

_

W

P r o fe s so r i n Ch a r g e

of

Th e s i s

TO T H E LIBRARIAN:------

1W TH IS T H E S IS IS NOT TO BE REGARDED AS CONFIDENTIAL.

GR AD . S C H O O L FO R M O—3 - 4 9 —1M

PiJltHi ACKNOWLEDGEMENT

The writer wishes to express his appreciation to all those among his family, friends, professors, stu­ dents, and colleagues who have waited patiently, and who have given him encouragement during the course of this investigation.

In particular, he is indebted to

Professor Irving W. Burr for the letter’s able guid­ ance and considerable assistance.

Acknowledgement is

also due to B, Tan Nostrand and Co, for their kind permission to excerpt certain material from the book by W. A. Shewhav-t, Economic Control of Quality of Manufactured Product.

TABLE OF CONTENTS

Page ABSTRACT 1.

2.

3.

..............................................

i

I N T R O D U C T I O N .......................................... 1 1.1.

Nature of the I n v e s t i g a t i o n .................... 1

1.2.

summary of Similar and Related Investigations .

1.5.

Previous Methods of A t t a c k .................... 4

1.4.

The

Chronology of the Present Thesis

.

.

.

A GEOMETRIC METHOD FOR FINDING THE DISTRIBUTION OF .

.

7 S

Definitions and Transformations

2.2.

The

General integral, b Infinite

2.3.

The

General Integral When b is Finite .

2.4.

Connection With the Distribution of Range .

.

54

2.5.

A Discrete Process Associated With the Method .

40

.

.

10

2.1.

.

.

2

.

10

.

19

.

21

.

APPLICATION OF THE METHOD TO VARIOUS POPULATIONS

.

46

5.1.

Normal P o p u l a t i o n ................................ 46

3.2.

Pearson Type III P o p u l a t i o n ....................47

5.2.1.

Jo int x ,s Method .

3.2.2.

Intervariate Range Method

3.3.

.

.

. .

. .

. .

. 4 7 .

56

Pearson Type I P o p u l a t i o n ........................ 59

5.5.1.

Rectangular Case .........................

62

5.3.2.

Right Triangular Case

3.4.

Prognosis of Further Results

.

.

.

.

.

64

.

.

.

.

.

65

Page 4.

SOME BY-PRODUCTS OFTHE M E T H O D ...................... 67

5.

S U M M A R Y ........................................... 72

APPENDIX.

SOME GRAPHSFOR

BIBLIOGRAPHY

p ( s ) .......................7/4

.......................................... 80

LIST OF FIGURES

Figure

Page

1.

Region of Integration,

n *3,

r*

space .

.

.

.

14

2.

Region of Integration,

n *4,

r*

Spaoe .

.

.

.

14

3.

Region of Integration,

n =3,

y*

space .

.

.

.

17

4.

Region of Integration,

n ■4,

y*

Space .

.

.

.

17

5.

Detail of Integration Region, n = 4, y* space

6.

Region of Integration,

7.

Detail of Integration Region,

8

.

9.

n *5,

y*

Space .

.

. .

n = 5, y* Space

Region of Integration for Range, n

=

. .

31

.

4, r space

summation Region in the T Matrix, n = 3

.

.

28

.

32

.

37

. 4 5

10.

Region of Integration, n s 3,

Sample Space .

.

.

49

11.

Region of Integration, n ■ 4,

sample Spaoe .

.

.

54

12.

Comparison of p{s), f(x) rectangular, with p(s), f(x) normal, n • 4

.

75

13.

Comparison of p(s), f(x) rectangular and triangular, with p(s), f(x) normal, n « 5 .................... 76

14.

Comparison of p(s) for the populations shown in Fig. 13, n ■ 2 ..................................

77

Comparison of p(s), f(x) Type III, with p(s), f(x) normal, n = 3 . . . . . . . . .

78

Comparison of p(s) for the populations shown in Fig. 15, n * 2 ........................ .

79

15. 16.

•

i

ABSTRACT

The main problem which is considered in this thesis is that of finding the exact sampling distribution of standard deviations when the sampled population is non-normal.

A

geometrical method is developed which solves this problem to within integration for all sample sizes when the sampled population has unlimited range.

In the case of a finite

range population, the required Integrals are given for sam­ ples of two, three, four, and most of the analysis is given for samples of five.

The necessary limits of integration

are supplied in each case. The method given represents a departure from the clas­ sical attacks on the problem in that (1) the fundamental geometrical space is taken to be the n-1 space in which the coordinates represent intervariate ranges of the sample variates, and (2) the fundamental region in this n-1 space is comprised of points corresponding to ordered samples and is consequently a

1/nl portion of the fundamental space.

essential steps in the development are as follows. ordered sample (xi,..^xn ) let n-1.

- x.^ = r^^O ,

For an i = 1,... ,

Then by the transformation with Jacobian, J,

find that the sample standard deviation, s, is given by •

( s *0' 1

=

%

S

•

ft? .

The

li

Then the fundamental region, U*, in the r* space is bounded by the hyperplanes and hypersphere p— — A.'w • ^ 7 7

^i.-»

» i'"1!*--) ^ ” 1

/*»-•

t ^

~

^ ’/y' s

1, .

If the underlying probability law in the population is f (x), a « x ,£ b t then the probability density for a point in the region U* is found by M-l r* S(n.*) * \ ^(x,)

Ax,

Ou

where the r^ are to be changed to r ^ r in accordance with the transformation given above.

When generalized polar coordi­

nates are employed, the density function is written and U designates the outer hyperspherical boundary of U T when ^ » JI5s.

Then the general integral for p(s), the prob­

ability law for s, becomes

(S)cAs

- /V»*

XT in which the coordinate angles may vary in accordance with

In the case of a finite range population with O ^ x ^ ^ k , the r* axes are orthogonally rotated until one of the new variables,

becomes the sample range, R.

It is shown

that the above integral, when adjusted for the rotation,

iii

will give p(s) for 0 i s < k/^/Sn, but that the form of p(s) changes as the hyperplane ynj,^ s k intersects U to give different shapes for this intersection. W

^

ft

T8"f

(/w1- IVij.

A*

There are

QVevi

for /M oJJ

connected arcs in the probability curve for s when the pop­ ulation has finite range. The method is used to derive nearly all known results for p(s), and a few new results are given.

Particular at­

tention has been paid to the Pearson Type XII and polynomial Type I populations.

Many more or less general and fundamen­

tal observations are made concerning the explicit functional form of p(s). It is of special interest to note that the distribution of sample range, p(R), is readily determined in the same fundamental region as is p(s).

The required integral for

p(R) is shown to be

/VI-3 if*,.3 where

1

is a linear expression in the trigonometric

functions of the coordinate angles of the region U, and where the limits of integration are the same as in the inte­ gral for obtaining p(s).

1 A GEOMETRIC STUDY OF THE EXACT SAMPLING DISTRIBUTION OF STANDARD DEVIATIONS WHEN THE SAMPLED POPULATION IS ARBITRARY

1.

1*1*

INTRODUCTION

Nature of the Investigation

The main problem to be considered in this thesis is that of finding the functional form of the exact sampling distribution of standard deviations when the sampled pop­ ulation is non-normal.

In general we shall deal with an

infinite continuous population wherein the underlying probability law shall be designated by f(x), and where the variate x may lie anywhere in the interval a ^ x ^ b * A sample shall consist of n independent observations chos­ en

from f(x) and is to have a sample mean, x, and sample

standard deviation, s, given respectively by

(1 )

The population mean, x, and population standard deviation, O’, are defined by

z To solve the problem we must, for a specified f(x), find the probability for s to lie in the interval ( s, s+ds) to within infinitesimals of higher order than ds.

We

shall consistently call this probability p(s)ds, and de­ note the cumulative probability function for s by P(s), where s (5)

-p (s )d ls

I Cs>

■ 1,2.

.

o

Summary of Similar and Related Investigations

The problem of finding the functional form of sampling distributions from non-normal populations has been found to be very difficult.

and Neyman and Pearson

gave the distribution of sample range for an arbitrary pop­ ulation and for all sample sizes.

Hall

and Irwin

found distributions for x when the populations were special cases of Pearson*s Type II.

Church

tion of x for the Type III population.

found the distribu­ The normal popula­

tion, however, is the only one for which we know p(s) for all n.

This distribution was first found by Helmert in

1876, deduced by "Student* in 1908, and later proved by R. A. Fisher in 1915. In the late 19S0*s and early 1930»s various writers found p(s) in some non-normal cases.

As an example of the

difficulty encountered we relate that R i d e r ^ f o u n d p(s) * All numbers in superscript brackets are for the refer* enoes listed in the bibliography at the end of this thesis.

3

for n=2, f(x) rectangular, and then hoped "that this re­ sult might offer some clue to the form of the distribu­ tion for larger values of n*.

He proposed the geometric

problem involved for n=3 in the AiaeriBan Mathemat ical Monthly.

Rietz solved this p r o b l e m t h e n found p(s) rcn for n=3 when f(x) is rectangular. A primary concern of most investigators has been to

find the effect of non-normality upon the "Student* tdistribution, and some results for p(s) arose more or less incidentally from the results of these studies. Geary ^

and B a r t l e t t ® h a v e given error terms for the

t-distribution when f(x) is two and three terms respec­ tively of the Gram-Charlier Type A population. lected terms in ^ any sample size.

They neg­

and n' although their results are for Laderman

gave the t-distribution

for n=2, f(x) arbitrary, and P e r l o ^ g a v e this distribu­ tion for n=3 when f(x) is rectangular.

Baker ^ h a s

ex­

hibited p(s) for n=2 and n=3, f(x) being three and two terms respectively of the Type A series.

He admitted

that "the general rule for the distribution of s is not apparent".

Baker ^

also found p(s) for n=5 when fix)

consists of two normal populations, and gave integrals for p(s) for higher values of n. shafei C41 gave p(s) for n=2 and n=3 for the Type III population, and Hsu has given an error term for the departure of pis), fix) arbi­ trary, from pis), fix) normal, when n is large.

In 1937

4

Rietz

gave an excellent summary of what was known about

the non-normal oases up to that date.

No new results seem

to have been given in recent years whereby it would appear that researchers have either considered the problem to be relatively unimportant, or that they have decided that the difficulties involved are more or less insurmountable. 1.3.

Previous Methods of Attack

There appear to be about five available methods for finding information about p(s).

The work of the present

thesis is closely allied to that of Craig and Truska. C r a i g ^ g a v e an analysis for the integrals which would lead to the joint distribution of x and s for n*4 with f(x) arbitrary.

In 1940 T r u s k a g e n e r a l i z e d Craig's method

for all n, obtaining the general integral in a recursion form.

Although the problem of finding the joint distribu­

tion for x,s was theoretically solved by this method, the fact that no new results were obtained indicates the great complexity of the integration required by this method. Furthermore, no analysis was given for finding the margin­ al distributions of x and s. Other writers have found p(s) by considering the geometrio requirement that the integration be performed over a hypercylinder in the sample spaoe.

This method is

closely related to Fisher's derivation of p(s) when the population is normal, and we have used it to some extent in this thesis.

5

In the third method the approach is to f i n d moments for pis) without regard for the functional f o r m of the distribution, these results being sufficient i n many pracf14.1 tioal applications. Craig ^ J * *- Jhas found expressions for the semi-invariants of p(s) in terms of t h e semi-in­ variants of f (x), his particular results b e i n g for the Type III population.

Kondo

apparently i m p r o v e d the

accuraoy of Craig's formulas, but experimental work under the direction of E. S. Pearson showed the f o r m u l a s to be in rather serious discrepance from observed r e s u l t s when n was less than 10 and/or

was greater th a n 0.5.

constructed a set of curves in

leRoujf16^

coordinates to show

the relationship between the moments of the s 2 distribu­ tion and the moments of the sampled population.

He then

could predict what Pearson types might p o s s i b l y represent the pis2) ourves if they could be found. Still another plan of attack is to deal w i t h a finite population which has a finite number of p o s s i b i l i t i e s for x.

Then all possible values for s can be c o m p u t e d and

the functional values for p(s) and P(s) can b e tabulated. The population frequencies for the x^ ean be varied to give various non-normal populations.

Rider £2 2 ^h.as used

this method to determine the t-distribution i n certain cases of non-normality.

Olds

Has also usecL it to find

the distribution of least and greatest v a r i a t e for a dis­ crete rectangular population.

Fr o m the p r a c t i o a l

6

viewpoint this met hoc5 should lead to results of any stipu­ lated precision sinoe it is fundamentally only a counting procedure.

With modern calculating equipment it seems

likely that a whole range of finite, discrete, non-normal populations could he studied in this way. Finally we observe that the bulk of research on the problem has taken the form of experimental sampling.

Two

of the more notable of these investigations were carried out by Church and "Sophister*.

Church

sampled two ex­

perimental populations using various sample sizes, recorded the s and s2 distributions, then fit the latter with Pear­ son system curves to find a satisfactory fit in most cases. Ee remarked that "many attempts were made to deduce without the method of moments the theoretical frequency curves of s2 for samples from the Pearson system".

"Sophister"

sampled from a Type IXX population for n=5 and n=2Q.

His

experimental data indicated that a Type VI curve was in order for p(s) so he determined the constants from the theo­ retical moments for p ( s 2 ), the latter being known for f(x) arbitrary, thus obtaining a **theoretical" curve for p(s). It would seem to us that "Sophister" was hoping for the success of "student" w h o assumed that p(s) would be Type III when f (x) was normal, ancL then was able to verify that such was the oase.

Dunlap

has experimented with dioe

throwing distributions for n -10 and concludes that "the distribution of s from a discrete rectangular population

7

is s k e w e d a n d leptokurtic". Ty p e

CheriyanP8-^whose f(x)

wsb

XII, a n d m a n y others have contributed experimental

r e s u l t s f o r p(s) In h i s

a n d p(s2 ) in various non-normal oases. rZd\ hook on q u a l i t y control, Shewhart Jgives experi­

m e n t a l sampling results for n=4 in the cases that f(x) is r e c t a n g u l a r and r i g h t triangular.

By permission of the

p u b l i s h e r s w e q u o t e from page 186 of his book.

"Theoret­

i c a l l y we k n o w n o t h i n g about the distribution function of the

s t a n d a r d d e v i a t i on of samples from a non-normal uni­

verse

- not even t h e values of the moments."

Although

thi s

statement is essentially true, it was slightly incor­

re c t

eve n at the time of its publication as is verified

by t h e

above summary.

We are also somewhat pleased to

r e m a r k that in t h e present thesis we are able to find the t h e o r e t i c a l c u r v e s for p(s) for his data, at least when the p o p u l a t i o n s a r e considered as being continuous. 1.4.

The Chronology of the Present Thesis

T h e r e w e r e f o u r rather clear-cut stages in the evolu­ tion

o f this thesis.

sample

At first we used the geometry of the

spaee to f i n d p(s) when f(x) was Type III, working

j o i n t l y w i t h x a n d s.

We obtained p(s) in general for

s a m p l e s of two a n d for particular values of The

when n=3.

i n t e g r a l for samples of four was set up but it ap­

p e a r e d that any ex a c t solution for this case was quite l i k e l y to b e uncomfortably close to the realm of mathemat­ ica l oddities.

S o m e weeks later we were further dismayed

8

to find in an overlooked reference that the Egyptian, 143 Shafei had already done the same thing for samples of two and three. We next started the "counting" method described above using a discrete rectangular population.

In time it be­

came apparent that the procedure was producing an under­ lying order which made it possible to pass to the contin­ uous population.

We could find the results given by

Rider ^ and Rietz^2"*and at the same time, and by the same method, we could find the distribution of range as given Il7l by Neyman and Pearson for a rectangular population; At this point we put aside the discrete method and decided that: (i) we may work with an ordered sample since the inde­ pendence of the

insures that the total proba­

bility for a sample is n! times that for the ordered sample; (ii)the intervariate ranges of the ordered sample can be regarded as the determiners of s, hence ignore x; (iii) we shall be operating in an n-1 space and may literally see the situation for n=4 with possibil­ ities for n=5. We then developed a theory far the determination of p(s) which appears to have advantages over previous methods and which leads to some quite general formulas and obser­ vations as well as to new results.

Although the theory

9

is relatively simple, it appears that no one has ever be­ fore used such an approach. The final stage has been to perform the integrations for various amenable populations.

We have not progressed

far with this phase of the problem, but the way seems clear, and we should be able to obtain new results for as long as we care to tackle new integrations. We suspect that there are Improvements for our method and it may well be that someone will discover a still better method.

For instance, perhaps there exists a

method analogous to the characteristic function method used in the determination of the distribution of sample means.

We reflect, however, that there are still wide

gaps in the solution of the latter problem even though, praotically speaking, we know all the essential informa­ tion about sample means.

10

2.

A GEOMETRIC METHOD FOR FINDING THE DISTRIBUTION OF 8 8.1.

Definitions and Transformations

We start with, a sample (x^, x2, ..., xn ) which is or­ dered, i.e.,

for 1=1, 2, ..., n-1.

We shall call

the non-negative distances between successive variates the intervariate ranges, r^. 16)

X l )

^

Then we have ,

O

1,2.,

i, =

.

. m.-\

.

Each x^ > x-j_ may he expressed in terms of x^ and the r^ by L-i

(7)

Xi =

X | -*■

22

A-j

,

i, = £.3,

.

i

Substituting (1) in (2) and using (7) we may write m. S £ — rrv.

£

«.= !

Xi' - ( £ X i f = „ f .

w- Z j= I

**=I

£ E *i + 1 ( s f } -

I

1=1 j - 1

1=)

j«V*i

>.=1

j= «

v=i j •=«.+»

J

ti-.i

I.S| jnti

Finally we have (8)

1 2 *•**

+ £, 2Z £

*»= »

i.0"-j)n.i>ai(

i-i

Now we wish, to "diagonalize" the quadratic form in (8) so that the hyperelllpsoid represented thereby will be a hypersphere in the transformed coordinates.

At the same

time we want the conditions represented by (6) to remain simple in form after the transformation.

After consider­

able experimentation we found that the following trans­ formation met our requirements.

Let

J

J

J

11

I.2-, ••'t The I n v e r s e of ‘this •transformation, and its Jacobian, J, are g i v e n by

(10)

From (6)

and (10) it is olear that the r^' are non-nega­

tive w h i l e

(6) and (9) together impose the conditions

If T i s the matrix of the transformation in (9) and M is the symmetric matrix of the quadratic form in (8) we find tliat the matrix multiplication TMT f= D yields n for E.

each e l e ment of the diagonal matrix D.

Hence (8) may be

w r i t t e n as

For r e a s o n s given in the next section, we shall be main­ ly c o n c e r n e d with the locus of points in the n-1 r* space which satisfy (12) and (IS) simultaneously.

Geometrically

these points must lie on the surface of an n-1 sphere with c e n t e r at the origin and radiusV£ns.

However, since

the r e g i o n defined b y (12) is but l/nl of the r ’ space, we m a y regard our required locus as being that portion of

IE

the n-1 sphere cut out by a solid "central" angle having no-I

radian measure 154).

/

11 /Vi! P ( /nE!)

(cf. Sommerville,^0^ page

We shall have occasion to consider the hyperplane

/vi- I

- R

, where R is the sample range.

From (9) we

L=M

find that (14)

R

r— -* V o(cff) l-l

+

Since we shall use polar coordinates in much of our devel­ opment, we shall change from rectangular to polar form by A, \

&er%> ip,

-

^ Saao. cp, CJOra ip (15)

ft\, ~

^

SLOvx

S A / w (^3

-

•

SaIoaj

f

«-z. “ ^ A*‘pa_, - ^

*' *

Sa/ww t^on-3 Cj°^t * SAo** l^/n-3 SUAO,

The Jacobian, J, of the transformation (15) is (15)

(«'2 , vX — ^ (sasvo

i

. /vi-3 »v»-M (5X00.^^ • • •

■ SA'VO, ^-s^'|/w-3

(13) we have as the conditions for our locus *?■ " (17)

•e.

"fOno>

(^3i 1 m*

"fa^

V^T7

2-

,

L--Z..S, ■■ .,/w-t

13

In polar coordinates (14) becomes tie)

R-nj; ^m-3 caot^n.^+

For simplicity we shall sometimes refer to the coefficient of

p

in (18) as

We shall use the letter U to

represent the locus of points on the hyperspherical por­ tion no matter what coordinate system we are using.

U*

shall designate the corresponding hyperspherical wedge. For samples of two, U is a single point on the r ^ f axis at distance

2s from the origin.

In Figs. 1 and Z we show

U and U* for n=3 and n=4 respectively in the r* space. The equations of the bounding surfaces are found in the margins of the figures.

In each figure the point marked

T is the point of tangency of (14) with (13). Whenever the population being studied has a finite range we find it convenient to transform the r^ in such a way that the sample range becomes one of the variables. To this end let (19)

Then (8) becomes (20)

14

Fig. 1. Region of Integration, n?3, r* space VUff: GUV: OTJW:

fi. = ^A.

ovw:

rv'i --f£n.3

Fig. S. Region of Integration, n*4, r* space

15

Now we seek a transformation on the

which will "diago-

nalize* the quadratic form in (20) as well as preserve y ,• n-1 We do not give any general transformation since we are not able in the present thesis to apply our method to a finite range population for samples larger than five.

Instead we

give the transformation matrices, P, for n = 3, 4, and 5, such that if N is the matrix of the quadratic form in (20), then PNP *= D , where again I) is diagonal with all elements n/2.

In matrix notation the transformation is [y^J =ryi,3 i>

where [y^l and [ y ^ a r e row matrices.

Actually these

transformations amount to no more than a rotation of the r* axes until the line 0T (cf. Figs. 1 and 2) is the y ^ ^ axis in the new system,

hence the Jacoblan of the trans­

formation is given by (11) in each case.

Now the inverse

of (19) is {

/t, = M i n‘L ~ V " V -

' i--*. 3,

•

So the conditions in (5) become the transforms of (24)

, -(ot-A> ±

(*-a) , ( ( ? | S S S * S S S "" s s s L

v >

y*

u a

\ J

({£, K < s < Zg) (67)

-pfe) - /viI \

•••

‘ jn-.-O x,-o~

V'°

Dldf f * e 3 r e r L t i a t i n g (70) with respect to R we have the probability

171)

l a w for sample range,

-j. C R j A T ^ m t A K

ck'* X,-Oj

IT

we

(72)

aad

use

(*

C "

^,-0

the notation

p (x.^l) = ^

i n t e g r a t e (71) successively with respect to the y^^ we

shall

have

36

r b-R (75) -pCR/>AT2

AK V

£ FC*.+*VF(*.^ $(*,) £(*,♦*>Ax, . X,-. CTo) *

^

(I")

■R - ‘2.

**■ y K

’ - R,*»

This brief discussion should be more clear after inspection of Fig. 9 which is a diagram of the situation as it exists in the matrix.

After the summation we replace R0 from (b) ,

and finally let TQ be 9s2 to have the cumulative probabil­ ity for s2 . Now suppose the population is changed to the fractions j/m, j ® 1,2,...,ink.

Letting m become large corresponds to

making finer and finer subdivisions in the matrix as it be­ comes denser with cell values.

In the limit, as m —*>oo ,

the sums above will be definite integrals in which terms of order 1/m, l/m2 , etc. will have disappeared.

In fact Fig. 9

would be Just half of Fig. 1 had we not transformed the r^ in the latter case.

it is clear that if we wish to study

a continuous population, nothing is gained by using the

45

Civc/edl ic-ffers «ve-Toy +he

case

F ig .

9.

process

of

H ow ever, lik e ly

if

th at

K-i ^ Ko R ®

S um m ation R e g io n th is

se ctio n

a d isc rete th is

If th e

th e

co m p u tin g

c ell

cre te

v a lu e s

is in

p o p u latio n .

added w ith p ro b a b ility

for

respect le v e l

th e

is

th at

p ro h ib itiv e ,

th e

can

th e

b ein g

n = 5 m ain m eth o d .

stu d ied ,

req u ired

th e

95^ le v e ls

T m atrix

Then

to

th e

a c o n tin u o u s

not

to

is

g iv e

found

99^o a n d

tw o d e c i m a l p l a c e s

T M a trix ,

preference

w ill

We h a v e

th e

th e

p o p u latio n

process

c o n cern in g p ( s ) . w ill d e te rm in e

in

in

P(s)

be

reached.

n = 5

found

T*s u n t i l

to

p o p u latio n .

p ro b a b ilitie s for

c e ll p ro b a b ilitie s

d ecreasin g

for

co rrect

rec tan g u lar th e

seem s

in fo rm a tio n

T m atrix

for

it

for

any d i s ­

may b e

any re q u ire d

46

3.

A P P L I C A T I O N OF THE METHOD TO VARIOUS POPULA TION S

..I.

Normal Population,

By ( 3 2 ) ,

and

f (x ) - yf^er 0

su b stitu tin g v **

(86; —

✓*1 V = ^~7 I

where

from

^ - «>