~--R66 364 A THEOREM FOR PHYSICISTS IN THE THEORY OF RAUO v/1YARIABLES RDDENDA(U) NAlVAL WEAPONS CENTER CHINA LAKE CAD T GILLESPIE JUL 83 NUC-TP-6462-RDD

UNCLSSIFIED F/6 12/1 NL

I EEEEEE...EEEEl

I lmllllllff

MlE CH

M'CROC~ CNAr

~IK~

( @ , ' QY ,. . ,- i , .', -. , • ''...'-I' -'.)- --.- .".,' ' 5"'_ .',-.

NWC TP 6462

€ . -. " -

Addenda to "A Theorem for Physicists inthe Theory of Random Variables"

by DTIC(0 Daniel T. Gillespie IELECTE

Research Department

~APR0 8I JULY 1983 0-

NAVAL WEAPONS CENTERCHINA LAKE, CALIFORNIA 93555

Approved for public release; distribution unlimited.

* mo-."

F-' .2

f 8 - .o

Naval Weapons Center* AN ACTIVITY OF THE NAVAL MATERIAL COMMAND'

FOREWORD

An article by D. T. Gillespie entitled "A Theorem for Physicists in the Theory of RandomVariables" was published in the American Journal of Physics, Vol. 51 (June 1983), pp. 520-33.That pedagogically oriented article called attention to a somewhat unorthodox prescription fordetermining the effects of subjecting random variables to mathematical transformations. Thearticle derived the "random variable transformation (RVT) theorem," and then illustrated theusefulness of that theorem by developing from it a number of important results in statistics andstatistical physics. Because of rather stringent page length restrictions in the American Journalof Physics (the current average length for an AJP article is only about four pages) severalappendixes, commentary paragraphs, and illustrative applications had to be deleted from theoriginal journal manuscript. This technical publication contains a compilation of the majoritems that were deleted, and therefore serves as a supplement to the journal article. This workwas done at the Naval Weapons Center during 1982 under Program Element 61 152N, Task

Area ZROOO-01-01, Work Unit 138070.

Approved by Under authority ofE. B. BOYCE, Head K. A. DICKERSONResearch Department Capt., U.S. Navy20 July 1983 Commander

Released byB. W. HAYSTechnical Director

NWC Technical Publication 6462

Published by..................................... Technical Information Department [Collation....................................................... Cover, 10 leavesFirst printing ............................................. 300 unnumbered copies

UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE Date __I_ __4_

REPORT DOCUMENTATIO4 PAGE BA coKm"PoI1. NROTUMER T .GVT RE CIPI T'S CATALON NUMBER

NWCTP "62 At Pt 033p4. TITLE (and Subtitle) S. TYPE OF REPORT A PERIOD COVERED

ADDENDA TO "A THEOREM FOR PHYSICISTS IN THE THEORY Supplement to journal article

OF RANDOM VARIABLES".. PERFomING oRG. RPOR NUuuui

7. AUTHOR(s) 4. CONTRACT OR @RANT N-UmbS*)

Daniel T. Gillespie

S. PERFORMING ORGANIZATION NAME AND ADDRESS I. PRQ P AMENTPROJ3CT. TASK

Naval Weapons Center 611 AIZ 4o,U I8UM 070

China Lake, CA 93555 152N, ZROOO0101, 138070

II. CONTROLLING OFFICE NAME AND ADDRESS IS. REPORT DATE

Naval Weapons Center July 1983China Lake, CA 93555 IS. PUcM OF PAGES

17

14. MONITORING AGENCY NAME 4 AO1RESSYI dlfferent from Controllif 011104) IS. SECURITY CLASS. (of NWe 0we4

UNCLASSIFIED

IS. DISTRIUTION STATEMENT (of tl Reporl)

Approved for public release; distribution unlimited.

17. OISTRIBUTION STATEMENT (of the abstract entered In Stock 0, It differeint Ibtm Repet)

IS. SUPPLEMENTARY NOTES

*iI

( A+ .-.

PC. KEY WORDS (Continue on reverse de It necessar aind identity by block number)

-Dirac delta function / -1' Random variablesGamma distribution Random variable transformation theorem • ,.-,

Normal distribution ; RVT theoremPoison distribution,)

20. ABSTRACT (Continue an revee slde It necessary end Identify by block namber)

See back of form.

AN gj EITION O NOV IS OSSOLTE UNCLASSIFIED

S/N 0102- LF- 014- 6601 SECURITY CLASSIFICATION OP T1IS PAGE (9ben DWth b'M9

UNCLASSIFIEDSaCURiTY CLASiFICA1tOft OF THIS PAGE (WAR D. 3II uu

(U) Aidenda to "A Theorem for Physicists in the Thory ofRandom Variables," by Daniel T. Gillespie. China Lake, Calif.,Naval Weapons Center, July 1983. 17 pp. (NWC TP 6462,publication UNCLASSIFIED.)

(U) This technical publication consists of ten addenda to thearticle -A Theorem for Physicists in the Theory of RandomVariables," which was written by the same author and published inthe American Journal of Physics, Vol. 51 (June 1983), pp. 520-33.The addenda are in the nature of comments, examples, andappendixes to the original work.

SE[CURIlTY CLASSIFICATION, OFr

THIS PAG(Wh"b~ DOIS I~EM006 % YJ

NWC TP 6462 ( , C)

This paper con often addenda to the article ,-kieorem for Physicists in the Theory of

Random Wriables r.c,' Journal ofPhysics 51 (June, 1983), pp.-520- 533t Numbered

references and equations referred to herein are those of that article.

ADDENDUM A. Obtaining Eq. (2).

Fourier's Integral Theorem states that, for any suitably behaved function Ax), we have

Ax) = (2n) 1 J ( ,,x')exp[is(x'-x) | dx'ds

If we put ftx)=6(x-x 0 ), then the x'-integration is easily performed, and the result is Eq. (2).

Students of quantum mechanics will recognize Eq. (2) as the orthonormality relation for the

eigenfunctions (2n)- 1/2exp(ixs) of the operator - i/as, with x representing the continuously

distributed eigenvalues.

ADDENDUM B. Derivation of Eq. (9).

Although the argument given in the text to deduce the integral formula for (h(X)) in Eq. (9)

from the definition of (h(X)) in Eq. (8) shows that Eq. (9) is indeed quite plausible, that argument

cannot be regarded as a rigorous mathematical derivation. In fact, a derivation that is at once

rigorous, general, transparent and brief does not seem to exist. As somewhat of a compromise, we

state and prove the following theorem, which is essentially a generalization of theio-called weak law

of large numbers. -

THEOREM : Let XP ..... X. be n independent random variables which have a common density

function P. Let h be any single-argument function for which the two integrals

m 0 h(x)P(x)dx (Bla)

and

exist. Then, for any given r >0,

PW M< I -B2)

Avaitblity CodesDist Avail adlar

1t Special

NWC TP 6462

Since we define h(X) to be the random variable whose sample values are h(xl1 ), h(X12) ..... whe r e P I ,

x(2 ) .... are the sample values of X, then the above theorem implies that the limit on the right side of

Eq. (8) "converges in probability" to the integral on the right side of Eq. (9). Notice that the theorem,

as stated here, requires the existence (i.e., finiteness) of the integral in Eq. (Bib). As one might

expect, the theorem is also valid even if the integral in Eq. (Bib) is undefined; however, the proof in

that case is much more complicated. [The more general theorem is due to Khintchine; see K. L.

Chung, A Course in Probability Theory, Academic Press (1974), pp. 109 and 169.1

To prove this theorem, we begin by formulating and proving a lemma that is essentially

a generalization of the well-know Chebyshev Inequality.

LEMMA I (Generalized Chebyshev Inequality): Let X - (XV ... , X) be a set of n random

variables (not necessarily independent) with joint density function Q. Let f be any function of

n real variables x - (,... .,x,,) for which the two integrals

M- 1 tx)Q(x)dx and S2. [Ax)-MIQ(x)dx (B3)

exist. Then, for any a> 0, we have

Prob{I /(X)-Mi - uS 15 aS-2 . (B4)

Proof of Lemma L: Let Dbe the set of all points x for which Ilx)-MI 1 aS. Then

I " I x ) -M 1 2 Q_ _ _ _ _ !Prob{IlX)-MI - aS}= Q(x)Jdx < I 1 Q(x)dxin xxx aS aS

=a-2 S- 2 [A(x)-M 2Q(x)dx = a

QED

Now we apply Lemma I to the case in which X1 ,..., X,, are n independent random variables

with the common density function P, whence Q(x 1 ... ,xn) = P()P(x2 ).-.P(x,,), and f is the function

. ... , ...,x W ! (1In that ... n) = - h(x.). (B5) $ ,.

In that case, the integrals M and S2 defined in Eq. (B3) are given by

M=f x 1 .. d~ ... '.h(x)jj P(x, (136a)-: ~ ~ 1= J = ,+,.+.and -= =

2-*z :- •, - .*. %t. ts ;..v,,- --

NWC TP 6462

s 2 = d.Jj, y &( )MAf Hp(.). (136b)j= 1

That M and S2 indeed exist is a consequence of the assumed existence of m and s2 in Eqs. (B ),

together with the following lemma.

LEMMA I: The quantities M and S2 in Eqs. (B6) are related to m and s2 in Eqs. (BI) by

M=m and S 2 =s 2/n. (B7)

ProofofLemma I: Letting all integrations run from - - to-, and using JP(x)dx = 1 together

with the definitions ofm and s2 in Eqs. (BI), we evaluate the integrals in Eqs. (B6) thusly:

M =1 n h(xi) P(x.) I dx h(x i) P(xi) = (nm) m.n J n

n2 Jdx1n I H P(le

i=tk1 ~6 %

- dx. dx _ h (x) h(x 2m h(xi) + nP(x2 J

~ J2 dx1 ... P(xj) + 2"- 1-• dx1 --- dxnh(x)h(x ) HP(xj)

rfl=! Fl F-1Rlei= ~

-2n.-'.,J dx,..,J dxh°,) [lP~x) + m, dx, ..-dx II,"<x )i= j=j=

2 dx h2(xi P(i) +- 2 --. --

n j=! j = klt{= J=i--"

- 2n-Im dx hxPx (x + 2.dx'P(x.l~| tI dx I I "1

fl ~ l z=1k4J=1

n 2 + ( -+dx h2(x) t) + ' IV mx h()- m2[

= - _ S...

n nn2n mf dxb2 xPx - ~ P~1 + mfPxh~)

F.F

2 2,

n n., 2_

_ ~ .~ M, .Z.xPWIh

NWC TP 6462

Substituting for f, M and S from Eqs. (B5) and (B7) into Eq. (B4), we get

Prob W dh(X)- m -asn - ta2 1 a - 2 . (B8)hi=1

Letting e = asn- 1 /2 , so that a- 2 = s 2 C- 2 n 1 , Eq. (B8) becomes, for any r > 0,

~ I 1g5CProb - " h(X) -m c n

or, since Prob{A <e}m I - Prob{A ae},

Prob lXJ(Xi)-m <c > l-s 2 C 2 (19

Now taking the limit of Eq. (B9) as n--*-, we obtain the desired result in Eq. (B2).

ADDENDUM C. Derivation of Eqs. (23).

The random variables K,, and Sn2 are defined in Eqs. (22) in terms of the random variables

X 1, . Xn" these in turn are defined in Eq. (21) in terms ofthe given random variable X. Since X1 ,

... X,, are mutually independent, each having the X density function P, then their joint density

function is P(x1 )P(x 2) ... P(xn). Thus, Eq. (18) implies that

(X)= dx... dxn xi [I P(Xk = x P(x,) dx, 1 J P(x)dx,- ] k=l - k=1 -

where the prime on the product denotes omission of k = i. Thus, using Eqs. (6) and (lOa),

(X) x Plx) dx= (X) u . (CI) -"

Similarly, it follows that

(X)= 2 x2P(x)dx (X 2). (C2)

Also. for any i *j, we have i"

(X X ... x ( x xj =~ x IP(x ) dx , x jP(x ) dx , ~ k) x- -"k=1 - -k=I

where the double prime on the product denotes omission of k i and k =j. Thus,

( XX? ( xI (x p2 = (i jl. (C3 .

4yV - '~\% v..:% V%'~'.% J% ~ :*.:~.* :%%~*.~~4*YX~.>~ ,*...-

NWC TP 6462

Using the above results and the linearity of the integration operation, we proceed as follows:

From Eqs. (22a) and (C1) we have

(n> -' 7_X,) = -' (Xi nx -' 7_ I,=1 =1 j=1

whence

ft (C4)

This establishes Eq. (23a). To prove Eqs. (23b) and (23c), we must first calculate (Z.2):

t= = =1i=1

21 2 2 YS 7 X) 2 -2 I'2n-~ ( )X+)n- n- n(X2 )+ n n (n-l~

1=11

where the last step uses Eqs. (C2) and (C3). With Eq. (10b), this reduces to

(Y?2) =n- 1 02 2.+ (C5)

Now using Eqs. (C5) and (C4), we get

(k 2) _ (X.)2 = n-'o 2 + 'p2 _ P2 = n-lo2,

which establishes Eq. (23b). Finally, we have from Eqs. (22b), (C2) and (C5),

S2 1 2- - ~ ~ 2S)=(n1 X X, n (X?)-x )

j=~1 j=1

= n(X2) - [- n -o 2 1 = (X2) - p2 _-no 2 .

Using Eq. (10b) this easily reduces to Eq. (23c).

If we define a new random variable _.2 by

S_2 =- (C6)

then Eq. (23c) evidently implies thatn 2) 02. (C7)

Some writers prefer to define the n-sample mean of Xtobe Z2 instead of S.2. Comparing Eqs. (22b)

and (C6), we see that as n-o- the sample values of both S, 2 and En2 approach 02; however,

comparing Eqs. (23c) and (C7), we see that only r,2 has 02 as its exact mean. Of course if n > t then

the distinction between S.2 and X.,2 is unimportant.

5.. ,,:..2:.h , *****

.- • ... . , . - . .. . • ". . " • ' % ,, ,,, . .. ",'• ; . %," ' ,, . . .,. . % k, % ., ' " . ,% % %

NWC TP 6462

ADDENDUM D. An Alternate Derivation of the RVT Theorem for n = 2. m = 1.

Let the random variables XI and X2 have the joint density function P(xl,x2), and let the

random variable Y be defined by Y = I(XtX 2). To calculate the density function Q(y) of Y, let us

assume that the equation y = AIxl ,x2) can be solved "nicely" for x2 in terms of x! and y; i.e., let us

assume that

y = ftxt,x 2 ) iff X2 = h(xly), (DI)

where the functions fand h are one-to-one with continuous first derivatives. Then the

transformation

T: (D2a)

Y = AXlX 2 )

from xtx2-space to xly-space is one-to-one and differentiable, and so is its inverse,

XI = xI

T- : (D2b) ,

Ix2 h(x1 ,y)

Let the infinitesimal area element dxldy be the image under T of the infinitesimal area

element dxldx2 . Then clearly {the probability that a simultaneous sampling of XI and Y will yield h

a point inside dxrdy I is identical to {the probability that a simultaneous sampling of X1 and X2will yield a point inside dxldX2 }. Letting R(x 1 y) denote the joint density function of XI and Y, we

can express this fact symbolically by a

R(x ,y) dxdy = P(x ,x 2) dxdX2 , (D3)

where it is understood that (xl,y) and (Xl,x2) are related by Eqs. (D2), and that dx~dx2 and dxldy

are related by the corresponding Jacobian formula

dx dx- = L2 I drdy = Ih Ix1 y)I dxdy. (D4)

Here, hy denotes the partial derivative of h with respect to y. Substituting Eqs. (D2b) and (D4) into

the right side ofEq. (D3), we find that the joint density function of XI and Y is

R(x,.y) = P(x,, h(x,,y)) Ih (x,y)l. (D5)

Therefore, according to Eq. (17), the density function of Y alone is

Q(Iy dx I Px I ,h(xyl))lIhv(x~y)l .(D6)

6

• -*

NWC TP 6462

Notice that this result has been obtained without reference to the Dirac delta function.

Now we observe that Eq. (D6) can also be writtenrI

Q(y) j dx! I d P(xrhlXr0)6h(Xl,4g16V-y). (D7)

Let us change integration variables in Eq. (D7) from (xj,0 to (xI, x2 h(,1k). Since x2 h(x1,4),

then it follows from Eq. (DI), that

=/( iVx2);

furthermore, just as in Eq. (D4) we have

ddx dx A2' 1h (x114)dx d4d1 2=

Therefore, Eq. (D7) becomes under this change of variable,

* Q(y) f idI dx2 P(xI,x2) 8(y-X 1 ,X2)), (D8)

which is precisely Eq. (31) for the case n = 2.

Although Eq. (D6) allows Q(y) to be calculated as a one-dimensional integral without having

to worry about any Dirac delta function, it describes only one of several possible ways of calculating

Q(y), and quite possibly not the easiest way. Eq. (D8) on the other hand offers a variety of ways to

proceed: If we change integration variables in Eq. (D8) from (xl,x2) to (x1 , 4a=fx 1 ,X2)) and then .

integrate over 4, we recover Eq. (D6); if we change integration variables in Eq. (D8) from (xI,x2) to

(-q =Il,x2), x2) and then integrate over q, we obtain a version of Eq. (D6) that would follow by

solving y = (Xl ,x2) for xt instead of x2; or, if we substitute into Eq. (D8) an explicit representation of

the delta function, suc, as that in Eq. (2), we obtain yet another avenue of colculation. Thus, Eq.

(D8) is not only more symmetrical, but also more flexible, than Eq. (D6).

Notice that the derivation of the general RVT Theorem given in Sec. Ill is much more

compact than the above derivation of the special case result Eq. (D8).

ADDENDUM E. The Delta Function Change-of-Variable Theorem. -- U,

The following theorem is useful when integrating over a delta function when the argument of

the delta function is a complicated function of the integration variable.

7...

NWC TP 6462

Theorem: Leth be a differentiable function of x whose only zeros are at x1, x2, ... xn, and let

h'(x) 0 for i=1.... ,n. Then

" 6(x-x.)(h(x)) Ih'(xI (El

Proof: Where fis any function of x, let

00 8(x-x) -

I- dx Ax) 6(h(x)) and J M- dx Ax) . (E2).... ,=t h'(x) A

The quantity J is easily evaluated as

n CO (x-x.) fix)d d- Ax) 7(xdx x) - (M)

( hilh'(x,)l

To evaluate I, let the x-axis be partitioned by points ao, al, an, which satisfy -- ao<al < ...

<an=-, and also xi((a i _ !,ai). Then

n

(E4)

where

I - dx fx) 6(h(x)). (E5a)

Now, since the argument of the delta function in Eq. (E5a) vanishes inside (ai- 1, ai) only at x =xi,

then we can also write Ii as

!. dx fix) 6(h(x)), (EMb)

where b, _ and b, are any numbers satisfying ai _ ! b, - I <x, < b a. Let us choose the interval

(b, - 1, bi) small enough so that h', which by hypothesis is nonzero at x = x, is nonzero everywhere in

(b, - 1, b,). In that case we can view the relation 4 = h(x) as a one-to-one mapping of the interval

(b, 1,, b,) about x =xi onto some interval (ci _, c,) about = 0. It is then permissible to change the

integration variable in Eq. (EMb) from x to =fix) by putting x = h- 1(4) and

d x - - dJd4/dxj lh'(x)[ jh'(h -1(Q){ '.

Eq. (E5b) thus transforms to

I' d4 fh '(0))! = d[, fAh- (4)) 61{) -, -;C, _ h'(h - (0 1 Ih'(h - '(0))l ,--

or, since x, is the only zero ofh in (b , b,),

8

* *-M

NWC TP 6462

(E54), h'(Xijt ,-

Inserting Eq. (E5c) into Eq. (E4) and then comparing with Eq. (E3), we conclude that l=J.

Therefore, we have shown that

Jdx x) 6(h()) d A:~ x) (EMx.I I(xA (_

for any function ft). Now taking fx) = 6(x- x'), the x-integrations on both sides are trivially

performed, and the result (apart from x being replaced by x') is Eq. (El). QED

As an sample application of the foregoing theorem, suppose h(x)=y-x 2 , where y>0.

Clearly, h(x) vanishes at, and only at, the two points xt =yl/ 2 and X2= _y1/ 2 . Also, since

h'(x) = - 2x, then Ih'(xi) = Ih'(x 2) = 2y1/2 . Therefore, Eq. (El) allows us to conclude that

& 2 0(yo x-y + S(x+yl) (E7&y-x) =(y) ,2(E7)2y 2

where the 0-function ensures that y is indeed non-negative (so that y1/2 is real).

ADDENDUM F. An Application of RVT Corollary Ii When f Is Two-to-One.

Suppose we wish to calculate the density function Q of Y = fX) = X2 when X is uniformly

distributed on [ - c1, c21, where ci and C2 are both non-negative. Since in this case f is not strictly

one-to-one (e.g., f sends both 2 and - 2 into 4), then Eqs. (39) are not applicable. But one-to-oneness

is not required of f by RVT Corollary 1I; thus, we can write from Eqs. (32) and (11),

Q(y)= _dx (ci+c 2 )- I O(x+c 1 )O(c 2 _x)l6(yx2). (FI)

Now, there are two ways of evaluating this integral. One way is to first render it trivial by

making the change of integration variable x--*z = x2 . The inverse of this transformation can be

either x = - z112 or x = + z112, but this is not an ambiguity so far as the above integral is concerned;

because, when that integral is written as a sum of two integrals, one over the negative x-axis and

one over the positive x-axis,0 D'

Q(Y)" dx (c,+c 2 )- 2O(x+c)6(y-x2) + 0 dx (cI+c 2 ) O(c2-x)6(y-x 2),

we have no choice but to take x = - zi 12 in the first integral and x = + z1 /2 in the second. Thus,

9 U

%.

NWC TP 6462

Q()= *(-z- In d4(c, +cl 9(-z 1 2+C1)6(Y-Z)

+ J O(+Jz 12dz)(c+c- 1 e(c2-ZM1 )6(y-z)

= [(c +C 1I'dzz- / 6(y-_Z) O(C _1 2) + O(C2 -z b)1

= ~ ~ d L2cc)j (z)r"z - Yz)[O(c2z) + O2cZ)I.

The z-integration is now trivially performed, and yields

Q(Y) = (Y)[6(C,-y) + (c2-y)J[2(c,+c 2)F 1y(F2)

Another way to evaluate the integral in Eq. (Fl) is to subject the delta function there to the

delta function change-of-variable theorem [see Addendum El. Thus, using Eq. (E7), we get

Q(Y) = J r d(c 1 + 2 I'6(X + CdO(C2- X)6() j a112 i~+M

I (W C2 2

= O(y)uMc + C2 )Jy i. I Y - /2 xrxc )O(C 2 x)6(X-y 1)

+ JdxO(x+c)O(c2 -x)6(x+y)I.2

Performing the x-integration gives

Q(y) = 0(y)[2(c +cq)F'y"2 {O(Y1 2 + +C1 O(C2 _y 12) + O(_yI/2+C1 ) 6(C2 +Y"2).

*Since y > 0, the quantity in braces reduces to {O(c 2 _ -y) + O(c1 -yfl, and we again obtain Eq. (F2).

10rp14j~ *J ~ **W~ ~ ~ * . ~ .****~~ *.* .. I

NWC TP 6462

ADDENDUM G. The Gamma Distribution.

If the random variables Xi ... , X. (n>- 1) are mutually independent and each Xi is E(a), then

the random variable Y defined by

Y = X9 (GI)i= 1

is said to have a gamma distribution with parameters a and n; we say that "Y is f(a,n)."

To calculate the density function of Y, we first observe from Eqs. (12) and (16) that the joint

density function of X1, ... , X, is

P(X19..... xn) O(Xi ) a exp(- axi)

iffi

Thus, by RVT Corollary I, the density function Q(y) of Y is

Q(Y) = dxl... dx,, [jO(xi)aexp(-ax) y- - x)j=

1

= a" I -1 exp(a I ) dx n O(x n)exp(-ax (x- n)- ' x, 1)i=t j=1

=O(Y)a" dx1 "".* dx -exp(-a '- x )0(y- x.)exp(-ay- X xj0 0 t=1 ==J=l1

Q(y) = O(y)a n exp(-ay) dx dx2 dx3--, dx 1 (G2)0 1 0 2 0 0G2

[The function O(y) was inserted when the delta function was integrated out because the argument of

that delta function never vanishes if y<O. Next we change integration variables from (xl ... x1n-)

to (Ul,...,Un.), where

U1 = y-X 1 , ,U2 = y-X 1 - X 2 ,

un- I = Y-Xl-X 2 - ... I-xnl1

Since this is a linear transformation whose matrix is triangular with - l's along its main diagonal,

then the absolute value of its Jacobian is unity; furthermore, since

j J-1

y- 'x y- x-x u -x .

>%.

u j Y , a -- ' X , X J - 1 - X .

NWC TP 6462

(with u0 -y), then as the old integration variable xj in Eq. (G2) ranges over the integral [0, ui_ 11, the

corresponding new integration variable uj ranges over that same interval. Eq. (G2) therefore

transforms to

fY "1 u n-3 U "n-2dQ(y) = O(y)a'exp(-ay) du 1 fidu2... dun_ 2 1 n- (G3)

This multiple integral is easily evaluated: The uq._ -integration gives ul._2/l! as the integrand

for the u-2-integration, which in turn gives u2 .- 312! as the integrand for the u.- 3-integration,....

which in turn gives ul'- 21(n- 2)! as the integrand for the ul-integration, which finally gives

yf- 1l(n- 1)!. We thus conclude that(ay)Y-

Q(y) = O(y)a exp(-ay) (-y- [Y is I(a,n)]. (G4),r (n- I )!

Using the integral identity Eq. (53), it is easy to prove from Eqs. (10) and (04) that the mean

and variance of the distribution I(a,n) are

(Y) = n/a and (Y2)_(y) 2 = n/a2 [Y is P(a,n)]. (G5)

It is obvious from the definition of the gamma distribution in Eq. (G) that

Fa,j) = E(a), (G6a)

which is also clear upon comparing Eq. (G4) with Eq. (12). However, it is a little surprising to

discover, by comparing Eq. (G4) with Eq. (54), that

P(j,n) = x2(2n). (G6b)

The asymptotic form of F(a,n) for n--a can be deduced most easily by using the Central Limit

Theorem: If Y is fRa,n), then from Eqs. (GI) and (22a) it follows that the random variable Z defined

by

n n - " ;

can be regarded as the n-sample mean of a random variable X, which in turn is E(a). Since X in that

case has mean a- 1 and variance a- 2, then the Central Limit Theorem tells us that the distributionI'

of Z tends to N(a- 1, a- 2 /n) as n tends to -. Thus, by Eq. (35), the distribution of Y= nZ tends to

N(na- 1, n2 a- 2/n), or

fla,n) -* N(n/a, n/a2 ) as n -- oo. (G7) *iEThe gamma distribution plays an obvious role in the mathematics of radioactive decay: Thus,

for a radioactive sample whose decay constant is r decays per unit time, the time required for exactly

12

NWC TP 6462

n decays is a random variable whose distribution is 11r, n). Eq. (G7) would then imply that the time

required for, say, 100 decays would be approximately normally distributed with mean 100/r and

standard deviation (100/r 2 )1 /2 = 10/r. Similarly, in a dilute gas with mean-free-path L, the total

distance travelled by a molecule in making exactly n collisions is a random variable whose

distribution is 1(1- 1, n).

ADDENDUM H. Sum of Independent Normal Random Variables.

Let X, ... , X,, be n mutually independent random variables, and let Xi be N(ui, oi2)

(i= 1...,n). Then by RVT Corollary 1, the density function Q(y) of the random variable

Y = XI (HI)

is

i=1 I~

Using Eq. (2) the delta function here can be written

Y I dsexp(isy) [exp(-is

j=f j= 1

With this, Eq. (H2) becomes

Q(Y) = (2)-T dsexp(isy) [1 dz. (2o 2)-1/2exp(-isx.)exp(-x.-ju ?2o?).W j=1

Changing integration variables from xj to zj= (xj -pjl/oj gives

nrQ(Y =(20 1 ds p's) [](2F-/ep-sj dzl exp(- io2exp(- z !/2).Q~~y)~ = 2Vs)i 2.

- j=1

Butdz. exp(- iso z.) exp(- z ?/2) = cos(so z)exp(-z 2 /2) dz= (20i/ 2 exp(- s2o F2 ),

where the last step follows from the integral identity Eq. (48). Thus,

Q(y) = (2n)- dsexp(isy) 1 exp(-isp .)exp(-shF2).

Now defining

1 3 4 1

.I "13

t >

t~ll i/" * i i" .°ili ("qP o i t *" "1

aI li" "l',l" i" " ! •.. . .. = ti~l - V .'.

NWC TP 6462

0 Y' and o "oj2 , (H3)-'j=

1 j=1

the above simplifies to

Q(y) = (20- 1 dsexp(isy)exp(-ispiexp(-s2 o2 /2)f-04 0

= (2n)- 12 Jos(s{y- 04Diexp(-s2ao12) ds.

Appealing once more to the integral identity Eq. (49), we conclude that

Q(y) = (2no02 )-/ 2 exp(-[y-p]0 2/2o0 2). (H4)

We have thus proved that the sum of n independent normally distributed random variables is itself

normally distributed, with mean and variance equal, respectively, to the sums of the means and

variances of the given random variables.

The foregoing result tells us that ifX, is N(juj, o12) and X2 is N(p 2 , 022), with X1 and X2

mutually independent, then the sum X1 + X2 is N(p1 + P2,o12 + 022). The difference X1 - X 2 is

handled by regarding it as the sum of X1 and - X2 : According to Eq. (36), -X 2 is N(( - t)p2,

(_ 1)2022) = N(-P2,02 2 ); thus, Xj-X 2 is N(g 1 -p2,01 2 +02 2).

If Xi is regarded as the ith sampling of X [see Eq. (21), where X is N(p,o2), then we have pt =,

and oj =o for all i = 1...n, and the foregoing result implies thatn

X is N(np, no 2).

i=! "

It then follows from Eqs. (22a) and (35) that, if X is N(p, 2), then Xn is N(n- l np, n - 2no2 )= N(p,

o2/n). This constitutes an independent derivation of the important result in Eq. (77a).

ADDENDUM I. Orthogonal Transformations

Following the classical exposition of Goldstein [see Ref. 71, we review here some properties of

orthogonal transformations that are invoked in Applications 8 and 10.

The linear transformation from Z12--.2n-space to ULiu2...un-space,

t

U = a. z (i= .... n), (Ill

is said to be orthogonal if and only if ".V

414

NWC TP 6462

,2= as ,2 (12)i=I

Substituting Eq. (II) into the right side of Eq. (12) gives

n n3 13 18

from which it is seen that Eq. (12) holds if and only if

Saa =6 (j,k ... n), (13)i.1k

where 6jk is the Kronecker delta symbol. This is one form of the orthogonality condition, although it

is not the form that we require in Applications 8 and 10.

Let the transformation inverse to Eq. (11) be

z s.= a* ik uk (j= 1,...n). (14)k =1 %

Inserting this into Eq. (11) gives

u= Y aia# jk Uk'

which can evidently hold for all ui only if

"a a' = 6 (i,k 1,...,n). (15)

Now consider the double sum,

Y a a a' 1 .i=1j=1

If we evaluate this by summing first over i with the help of Eq. (13), the result is a'ki. But if we sum

first overj with the help of Eq. (15), the result is alk. We conclude that,.

a k (k,l .... n). (16)

Substituting this into Eq. (15) gives

a ak = 6 k (i,k=l,...,n), (17)i

i= I

which is the form of the orthogonality condition invoked in Eqs. (69) and (107a).

15•-,,..- '

NWC TP 6462

The Jacobian of the transformation in Eq. (II) is clearly equal to the determinant of the

matrix A, where (A)ii-au:

(ul ... ,u ) (18)d(z I9 .... I z t)

Denoting the inverse of A by A- 1 and the transpose of A by AT, we have

I = JA - I JA ATI = A IATI - [A 2 • (19) .

Here, the second equality follows from Eq. (16), the third equality follows from the fact that the

multiplication rule for matrices is the same as the multiplication rule for determinants, and the last

equality follows from the fact that transposition does not alter the value of a determinant.

Combining Eqs. (18) and (19) gives

"zl. I I. (110) € €a3(u1,.... ,u)

Eq. (12) says that orthogonal transformations preserve length, and Eq. (110) says that orthogonaltransformations preserve volume.

ADDENDUM J. Asymptotic Equivalence of the Poisson and Normal Distributions

Using an argument similar to that given by Present [see Ref. 91, we show here that the

Poisson distribution with a large mean can be approximated by the normal distribution that has the

same mean and variance. This fact was invoked in our proof of the chi-square theorem [cf. Eqs. (95)

and (96)]. .-.

The discrete-variable Poisson probability function with mean and variance a is

P(x) = aXe-a/x! (x 0, 1,...). (Ji)

We consider here only the circumstance

a > 1, (J2)

in which case P(x) differs appreciably from zero only if x satisfies

1x-al <a. (3

If a satisfies Eq. (J2), then any x satisfying Eq. (J3) will be very large compared to unity; we can

therefore invoke Stirling's formula,

xA 1 (2nx)6xe - ' (x > 1), 04)

16*

NWC TP 6462

a n d so a p p ro x im a te E q . (J ) b y -2e

P(x) (2nx) -I 2 (a/x)X e-a

= (2na) 1/ (Xia) -(X+ 1/2)

- (2na) - 2 W(x/a) - z ex- a ,

where the last step follows since x), 1. Rearranging and taking logarithms gives

log[(2na) " 2 P(x)j -x log(xla) + (x-a),

or, defining e - x-a,

log[(2ra)" 2 P(x)WI -(e+a) logil + (la)I + e.

Eq. (J3) implies that fr/al 4 1, so we can expand the logarithm on the right and retain only the lowest

order terms in r/a:

logl(2na)1 /2 P(x)] - (e + a) [ (s/a) - I(ela)2 + . + V N-

C --- (e2 /a) - -(x-a) 2 /2a.

We therefore conclue that, when a> 1, the Poisson function in Eq. (Ji) can be approximated by

P(x) - (2a) - 112 exp[ - (x- a)2/2a] (J5)

in the region where it differs appreciably from zero. The quantity on the right of Eq. (J5) is of course

the density function of the normal distribution with mean and variance a.

S4.

.%.

17-'e

4.

f

s-U

1i~ 4 -. ~- ~*~*:M- ~ *-*