The Expected Value and Variance of the Reciprocal and Other Negative Powers of a Positive
Bernoullian Variate
Author(s): Frederick F. Stephan
Reviewed work(s):
Source: The Annals of Mathematical Statistics, Vol. 16, No. 1 (Mar., 1945), pp. 50-61
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2236177 .
Accessed: 20/02/2012 10:57
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The
Annals of Mathematical Statistics.
http://www.jstor.org
THE EXPECTEDVALUEAND VARIANCEOF THE RECIPROCALAND
OTHER NEGATIVEPOWERS OF A POSITIVE BERNOULLIAN
VARIATE'
BY
FREDERICK
F.
STEPHAN
WarProductionBoard, Washington
1. Introduction. The expected value of the reciprocal of a Bernoullian
variate appearsin certain problemsof randomsamplingwhereinboth practical
considerationsand mathematical necessity make zero an inadmissible value
of the variate. This specialconditionexcludingzerois necessaryfroma practical
standpointbecausestatistics can not be calculatedfrom an empty class. It is a
necessary condition, in the mathematical sense, for the expected value, and
variancesinvolvingit, to be finite. When subject to this conditionthe Bernoullian variate will be designated the positive Bernoullian variate.
There appearsto be no simple expressionfor the expectedvalue of the reciprocal such as there is for the expected value of positive integral powers of the
positive Bernoullian variate. This paper presents in (15) a factorial series,
which can be computedconvenientlyto any desirednumberof terms by means
of the recursionrelation (18). Upper and lower bounds on the remaindermay
be computedreadilyfrom (20), (21), (23), (24), and (26) and the approximation
may be improvedby adding an estimate of the remaindertaken between these
bounds. A factorial series for the expected value of negative integral powers
is given in (34). A factorialseriesfor the expectedvalue of the reciprocalof the
positive hypergeometricvariate is given in (53). Seriesfor the variancesfollow
directly from the series for expected values.
A simple example of the sampling problems in which this expected value
appearsis presentedby the followinginstanceof estimatesderivedfrom samples
of variablesize:
An infinite population consists of items of two kinds or classes, A and P.
Lots of N items each are drawnat random. In such lots the numberof items,
x', that are of class A is an ordinary Bernoullianvariate. Next, every lot
composedentirely of items of class B is discarded. This excludes all lots for
which x' = 0. From each remaininglot the N - x' items of class B are set
aside, leaving a sample composedentirely of items of class A. The numberof
such items, x, varies from sample to sample. It will be designateda positive
Bernoullianvariate since x = x' if x' > 0 and x does not exist if x' < 0. Finally,
let there be associatedwith each item in class A a particularvalue of a variable,
y, the varianceof which in A is a2. Then if the mean value of y is computedfor
each sample, the errorvariance of such means is E(u2/x) = cr2E(1/x).
Instancessimilarto that just describedoccurin the designof samplingsurveys
from which statistics are to be obtained separately for each of several classes
1Developed from a section of a paper presented to the Washington meeting of
the Institute of Mathematical Statistics on June 18, 1943.
50
BERNOULLIAN
51
VARIATE
of the population, i.e., each statistic is to be computed from some part of the
sample instead of all of it. They also occur in certain sampling problemsin
which some of the items drawn for a sample turn out to be blanks.
A related problem concerningthe error variance of the proportionof males
among infants born in any one year was consideredby G. Bohlmannin a paper
on approximationsto the expected value and standard error of a function [1].
His approachto the problemwas to expand the function in a Taylor series and
take the expectedvalue of each term. The conditionsunder which the resulting
series converges were developed for certain functions of a Bernoullianvariate.
The presentpaperprovidesa differentand, in certainrespects,superiorapproach
to the problem employing a method due to Stirling [2]. While the method is
applied to the reciprocaland negative powers it is also applicable to certain
other functions of a Bernoullianyariate.
2. The positive Bernoullianvariate. Let x be a randomvariate definedby a
Bernoullianprobabilityfunction subject to the special condition x > 0. The
probability of x in n is
(1)
P(x)
=
(U)
p
-
qn'-/(1
q n)
wherex and n are integers,1 < x < n, and
(2)=
_
(2)
_
_
(~~~~~x)x!(n- x)!'
The probabilities p and q are constants, 0 < p = 1 - q < 1.
The divisor 1 - qn arises from the condition excluding zero. (Bohlmann
omits this factor, assuming that qn is negligible, an assumption that is not
n
An extension of this condition to exclude
always valid. In fact, qfl &.)
all values of x less than a specifiedconstant will be consideredin a later section.
Throughoutthis paper summationis understoodto be from x = 1 to x = n
unless it is shown otherwise.
3. Expected values and moments. The expected values of x and its positive
integral powersare
E(x)
(3)
(4)
= np/(l
E(x2) = (npq +
-
n2p2)/(l
qn)
-qf)
and, in general
(5)
E(xt) = vi/(l - qn) =
n
>
0
where vi is the ith moment about zero of an ordinaryBernoullianvariate with
the same n and p and the S' are the Stirling numbersof the second kind (see
Table 1).
The moments about E(x) are somewhat more complicated than the corre-
52
FREDERICK
F. STEPHAN
sponding moments of the ordinary Bernoullian variate. For example, the
variance
(6)
EI(x
pq
=
E(x))2Jn21
-
nq"
P
(1-qX)p
n_
-q
and the third moment
1
q (1 + qn)
_
p+ n
E(x))3} = (q p)npq
(1 - qn)3
(1 - qn)2
1 - qf
The moments about np, the first moment of an ordinaryBernoullianvariate,
are
(7)
El{(z-
Et(x - np)') = (usi + (-1)'N'(np)qn)/(1
(8)
qn)
-
TABLE 1
Stirling numbers of the second kind, S'
-
4
3
2
1
6
5
1
1
0
0
0
0
0
2
1
1
0
0
0
0
3
1
3
1
0
0
0
4
1
7
6
1
0
0
5
6
7
8
9
10
1
1
1
1
1
1
15
31
63
127
255
511
25
90
301
966
3,025
9,330
10
65
350
1,709
7,770
34,105
1
15
140
1,050
6,951
42,525
0
1
21
266
2,646
22,827
where g,iis the ith moment, about the mean, of an ordinaryBernoullianvariate
with the same values n and p.
The expected value of the reciprocalis
E(1)
=
1
q {1npq nTh+
(9)
2
n(n
-
1)p2qn-2
1(
-
+
+
n
This equation is not suitable for the computationof E(1/x) to a satisfactory
degreeof approximationunless np is small, say less than 5 for most purposes.
The numberof terms necessaryto obtain a computedvalue with four significant
figures,for example, may be estimated to be approximately8,\/npq/( - qn).
Expressedas a function of q, E(1/x) becomes
(10)
EQ n)
+ 1
a serieswhich may be convenientfor small values of q.
E(1/x) may be expanded in a power series by Taylor's Theorem. It may
BERNOULLIAN
53
VARIATE
also be expandedin a finite series of expected values of powers, either in E(x),
E (X2),
_.
or in E(x
-
c), E(x
-
c)2, *-- c being any positive constant.
secondof these three seriesmay be obtainedby expanding 1
1
1
expected values, and the third by dividing out 1 =
1
+
The
and taking
-
)and taking ex-
pected values. For all three expansions,however, the terms become progressively more complicatedand laboriousto compute. A simpler and more convenient series for actual computationsmay be obtained by expanding l/x in a
factorial series.
4. Expansionof E(1/x) in a series of inverse factorials. It is easy to prove
by induction that, x > 0,
1
0!
x
x +1
1!
(i-l
(x + 1)(x + 2)
!x
(x+i)!
+
+ (t -1)!x!
...
(x + t)!
where
Rt(x) = t!(x
(12)
+ R (x)
1)!/(x + t)!
-
is the remainderafter the first t terms. This is, of course, an expansion in
Beta functions. It is also a simple special case of the expansionof a function
in a "faculty series" or series of inverse factorials [3] with an exact expression
for the remainder.
Let
(13) si = 2; (+
Then, since
=
i P+
(14)
X!
(x + 1)
the expected value of (11) is
(15)
E(1
\~xJ
-
O!s,
(n +1 )p
+
1-q (1
(n\ pq
-
n+i
.
n!s; (1
qn)
-
p
+
%x/(
pz qn+i-)
+ (i - )!n!si
1!s2
+
(n + l)(n + 2)p2
(n + l)!pi
+ (t 1)! n! St + 2Rt(x)P(x).
(n + t) !p'
When developed as infinite series, both (11) and (15) are convergent since the
remainders Rj(x) -O 0 as t -* o.
For computingpurposesit is convenientto write
+
(16)
E(l)
= E ui + E(Rt(x))
in which, since
(17)
=
-q(j&+
'
i
n-1
54
FREDERICK
F. STEPHAN
the following recursion relation exists between ui and ui-i
(18)
(i - 1)!n! si
(i - l)u,
-kli >1/
(n +i)p
+i)! pi
~~~(n
(18)
i
=l
(n + 1)p
where
k = npq7/(l - q )
1).
np/(e
(19)
This reduces the computing of the ui to a simple repetitive procedure. The
computing is still simpler in those problems in which, for the degree of precision
desired, k is negligible.
An estimate of E(RL(x)) should be added to the sum in (16) to improve the
approximation. To determine a suitable estimate, a lo4ver bound for the expected value of the remainders may be computed from one of the following
inequalities:
E(Rt(x)) = 2;t (t -1)!x!P1(x)
x (x + t)!
(20)
= 2:t(
>
m
m + (z-m
_ x
1 t(t -l)ut+
tut -
(21)
m+ttue,
E(Rj(x)) >
{(t -
1)tt- -tut}/ut
_)Ut_l
tU2/1(t-
)
m?0
m
m
whichis maximizedby setting m =
)' (x + )!x1(
,
whence
t > 1.
_tUil,
Also, since when m = E(x)
(22)
(t -1)!
3(x-m)
P(x)
(x + t)!
< 2(x
-
m)P(x)
=
0,
a simpler inequality is
(23)
E(Rt(x)) > tu1(l - q')/np.
Further, if only the first c < n terms in (20) are taken,
t! (x - l)P()
E(Rt(x)) >
(24)
(x + )!v
E
where
(x-1)(n-x+1)
x(x + t)q
qIA.
from
be
An upper bound may
computed
tug
(26.1)
1tu + 2
(26.2)
(25)
(25)
(26)
VI vl
(
)
and
(t + I)q
E(Rt(x)) <
1
tug
1
2
vz
2
1
+-V1+-6V2
+ E
XI
\XJ/
(26.3)
(26.j)
55
BERNOULLIAN VARIATE
the choice among which may be governed by computing convenience.
With (16), these inequalities provide lower and upper bounds for E(1/x).
Taken
Two examples will serve to illustrate the factorial series (15)..
5. Examples.
EXAMPLE 1
Computation of E(l/x) for n = 100 and p = 0.1
np = 10
k = .000,265,621
E(1) = .111,527
t
Binomial
sum of t
terms
Sum of t
terms
Factorial
series lower
bounds*
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
.000,295
.001,107
.003,071
.007,039
.013,813
.023,743
.036,442
.050,796
.065,287
.078,474
.089,372
.097,604
.103,320
.106,985
.109,164
.110,369
.110,992
.111,294
.111,431
.111,489
.098,984
.108,675
.110,548
.111,082
.111,280
.111,370
.111,416
.111,444
.111,461
.111,472
.111,481
.111,487
.111,492
.111,495
.111,498
.111,501
.111,503
.111,505
.111,506
.111,508
.099,647
.109,006
.110,752
.111,223
.111,385
.111,452
.111,483
.111,500
.111,509
.111,514
.111,518
.111,520
.111,521
.111,523
.111,524
.111,524
.111,525
.111,525,4
.111,525,6
.111,525-8
24
.111,526
100
Upper
bound**
.132,167
.115,247
.112,498
.111,852
.111,657
.111,587
.111,556
.111,544
.111,537
.111,534
.111,532
.111,530
.111,529
.111,529
.111,529
.111,528
.111,528
.111,527,5
.111,527,3
.111,527,1
(.111,034)
(.111,313)
(.111,381)
(.111,452)
(.111,478)
(.111,489)
(.111,497)
(.111,503)
(.111,508)
(.111,511)
.111,527 (end of series)
* Sum of t terms plus lower bound for E(R(x)) from (24) with c
bers in parentheses are calculated from (21).
** Sum of t terms plus upper bound on E(R(x)) from (26.3).
=
3.
Num-
EXAMPLE 2
Computation of E(1/x) for n = 1000 and p = 0.3
np = 300
t
Sum of t terms
1
.003,330,003,330
2
.003,341,081,185
*Computed as in Example 1.
k = 9.7 X 10-14
Factorial series upper and lower bounds*
f .003,346,7
.003,341,0
(.003,341,155,4)
56
FREDERICK F. STEPHAN
t
Sum of t terms
3
.003 ,341,154,817
Factorial series upper and lower bounds*
.:003,341,155
4
4
~~~~~~~~1.003,341,156,2
.003,341,155,549
f.003,341,155,56
{55
.003,341,155,55934
5
5
f.003,341,155,58
.003,341,155,559
For the binomial series, the sum of the largest eight terms of (9), not the
first eight terms, is approximately .0007 which is less than 1/4 of the
value of E(1/x).
In.the first examplethe value of np is almost small enoughto make computation
by (9) convenient. In the second exampleabout 120 teims of (9) must be computed to obtain an approximationto four significantfiguresbut only four terms
of the factorial series are needed to obtain seven significantfigures. It is evident that as np increases,the number of terms of (16) requiredto obtain an
approximationto a given numberof significantfiguresdecreases. The opposite
is true of (9) as n increases,or as p approachesa value near 1/2.
6. Extending the special condition. In some sampling problemsall values
of x less than a specifiedvalue, g, and greater than another specifiedvalue, h,
are inadmissible. Then the probability of x in n is
P(z Ig, h) = ( Pn
(27)
g.
4q`1-9/8
Z < h,
g<
where
= LI
SO,g,h
SO.gP,h
(28)
Ipq
t
zx-g
With this new condition, E(1/x) is given by (15) if si is replacedby
(29)
=4
+(~
X=o(Z
+
:
PX-q
s)
X.g
and the summationin the remainderterm is from g to h. Also since
85.u.h
(30)
= Si-1.e.h-
q
8O,UJ4
+
l+
q,'-0
-1)u+i-.
g~~~~~~~~~ + i
-
n~;j1
h+iqfl-h-1}
57
BERNOULLIAN VARIATE
a recursion relation similar to (18) may be used in computing
(i -l)n!si,,h,
=
(31)
s =
(n
(i
+ i)! pi
-
(i
-
)u
where
+ i (n + i)p
1)! {k/(g
-
1)! + kh+l/(h + i)}
n!pgqn-g+1
(32)
k=
(n g)!!so
n! ph qn -h+1
(33)
- h) ! 8og,
~~~~~~~~~~(n
The inequalities (20) to (23) inclusive and (26) are applicable to this extension
on substitution of Ui,g,h for u,.
7. Expansion of E(x-a) in a factorial series. Equation (11) may be extended
to other negative integral powers of x. If a is a positive integer
(34)
E(x-a)
=
2
1 P -()
Si
=
XIP(' -(n+I)
p
+
b2a+
(n +l1)(n
+ 2)2)p2
P(x)
ZR (z)P
+ + .. + ( t gs ! g+ -zR
where
(35)
R'(x) =
b,+,.j X
x!(x
t)! ,
and the bi,j are the absolute values of the Stirling numbers of the first kind (see
Table 2) formed by the recursion relation
(36) bi,j = b-1,,1 + (i
-
bi,j = 0 if j > i or j < 1.
)bi-,, ,
It is evident that
(37)
(38)
j-1
bi, = (i-
bi,j = i!
1)! and bi, ,< i! if j > 1,
whence
R'(1)
(39)
=
1 (1)
t+1
R (x) < ((t + )! - t!)x!P(x)
2(x+).
<P(x).
(t +1)
x>
1
58
FREDERICK
F. STEPHAN
Hence R (x) -+0 and E(R,(x)) -+0 as t
(34) converges to E(x-a) as t
oo and the sum of the first t terms of
- oo
The followingrecursionrelationcorrespondingto (18) providesa simpleprocedure for computing:
(40)
Ui,a = bi.aUi/(i
= bi,a ( ni-+)a/bp-ja)-kli!
-1)!
The computing procedure,then, follows a cycle of four simple operations:
1. Divide {k/(i -
1)!I by i.
2. Subtractthe quotient from {ui-1.a/b,_1,a}.
3. Divide the differenceby I(n + i + 1)p} + p. The quotient is Ui,a/bia.
4. Multiply this quotientby bi,a.
TABLE 2
Absolute values of Stirling numbers of the first kind, b, ,'
1l
1
1
2
0
1
3
5
4
3
6
4
6
11
6
1
5
24
50
35
10
225
85
15
1
1,624
175
735
13,132
6,769
1,960
22,449
118,124 67,284
1,172,700 723,680 269,325
21
322
4,536
63,273
6
120
274
7
8
9
10
720
5,040
40,320
362,880
1,764
13,068
109,584
1,026,576
0
0
0
0
0
0
0
0
0
0
0
0
1
1
2
3
*
2
0
0
1
These numbersare also known as differentialcoefficientsof zero [4].
The expressionsin braces are quantities obtained in the precedingcycle.
The ui,4may also be calculatedfrom (18), or checkedby such a calculation.
A lower bound for E(R'(x)) after t terms may be calculatedfrom the first c
terms of
E(R'(x)) = S Rt(x)P(x) >
z-1
(41)
R
R(x)P(x)
_
(
Ea
:-X -1jl
ei+l(x
or from an inequalitysimilarto (23)
(42)
E(RJ(x)) > (tt1- 1)!! _E_____
i-i
Ex)-+
bg+iJn!pxq
+ t)! (n
-
x)! (1-q)
BERNOULLIAN
59
VARIATE
which may also be written
ER(x))
> (t(t
(E(x) + t)(E(z) + t-1)
a
1)
b-
,(E(x)4Ex.
... E(x)
-
An upper bound may be calculatedfrom
(44)
E(R'(x)) <
(t
Ut
-
I!
1)!
bt+,,i < t(t + 1)ut
or
E(R'(x)) <~
(45)
R'(x)P(x) + ~
3b,+,1, (x
+ t)! +
-C+L1
ut
< E R'(x)P(x) +
=
b'+l.i
(t - 1)! j-1
3R'(x)P(x)
+ (t _ 1)I +1{(c + t)(c + t - 1) ***c-
j21b+.1c'}.
8. The positive hypergeometricvariate. The theory of sampling without
replacementfrom a finite populationrests on the hypergeometricvariate. Its
probability function is
(46)
P(x IN, M, n) = (M)(N
M
In applicationsto finite sampling,N is the numberof items in the population,
M is the numberof them that are of a certain kind, n is the numberof items
drawnfor the sample,and x is the numberof items of the designatedkind in the
sample.
As in the case of the Bernoullianvariate, it is necessary to exclude zero in
defining the expected value of l/x. The probability function of the positive
hypergeometricvariate, then, is
(47)
PH(x) = P(x I N, M, n)/so,
x > 0
where
(48)
8o
=
1
-
P(O I N, M, n).
Throughoutthis section the notation will have referenceto (47) instead of (1).
The expected values of positive integral powers of x are
E(x) = Mn/(Nso)
(49)
(50)
E(x2) = ] {M NMl)n(n-1)+
60
FREDERICK F. STEPHAN
and, in general,
(51)
6' E(x!(x-j)!)
E(xS) =
,1=~
where the ' are the Stirling numbersof the second kind and
(52)
E(~~X!
~~M!n(N
-j!
x -j) !
(M-j)! (n-j)! N!so
The factorial series correspondingto (16) is
1
(53)
E(1)
?
=
u,+
PH(x) =
E(Rt(x))
where
(54)
-
=
- l)x! P(X)
(x + i)
and
E(Rt(x)) = 2;t! (x-)
(x +t)
The ui may be computed from
(55)
PH(X)-
-
Ul
(56)
=
(N +
1)si
(M + 1)(n + 1)8o
1
N+ 1
(N-M)!(N-n)!
so (M + 1)(n + 1)
and the recursionrelation
(57)
N!(N
M
-
-
n
-
1)!(M + 1)(n
1)
+ i)8i
~~~~~~(N
(M + i) (n + i)sj1 i-i
where
(58)
8i=
1-FP(xjN+i,M+i,n+i).
xzO
The computingis quite simple in those instances in which 1 - st is negligible.
Correspondingto (26), an upper bound for the expected value of the remaindersafter t terms may be computedfrom
(59.1)
tug
(59)
Itut + IPH(1)/(t + 1)
2
PH(1) +
(2)
E(Rt(x)) < itut + 3t+
1 6 (t + 1)(t + t)
-tUgt!
X
-
-)
P,,(x) (x +
)F(59-j)
(59.2)
((593)
61
BERNOULLIAN VARIATE
A lower bound for the expected value of the remaindersmay be computed
from one of the followinginequalitiescorrespondingto (23), (21) and (24)
(60)
E(Rt(x)) > tutNsol(Mn)
(61)
E(Rt(x)) > tu2/{(t-
-tud
t H(X)
X- (x + t)!
The expected values of other negative integral powers of the positive hypergeometricvariate may be calculatedfrom
E(Rt(x)) >
(62)
E(x-a) =
(63)
i, 1
bi,aui/(i
-
1)! + E(Rl(x))
where
=
R(x)
R'(X)
(64)
a
xi''iX!PH(X)
~bt+II'jxa(x +t)
P$
With PH(x) substituted for P(x), (39), (42), (43), (44), and (45) provide lower
and upper bounds for E(Rt(x)) for the positive hypergeometricvariate. Also,
correspondingto (41)
(65)
E(R,(x)) > E Rt(x)PH(x).
xl-
9. Variance and moments of llx and x-a. The variance of 1/x, which is
E(1/x2) - (E(l/x))2, may be calculatedfrom (16) and (34), with.a = 2, for the
positive Bernoullianvariate, and from (53) and (63), with a = 2, for the positive
hypergeometricvariate. Likewise, the variance of x-a and the moments of
llx and x-a about E(1/x) may be computed by the usual formulae.
REFERENCES
"Formulierung und begriindung zweier hilfs8atze der mathematische
Statistik," Mlath.Annalen, 74(1913), 341-409.
[2] E. T. WHITTAKER AND G. ROBINSON, The Catculu8 of Observations, London (Second
Ed.) 1937, p. 368.
ModernAnalysis, Cambridge (Fourth Ed.) 1927,
[31 E. T. WHITTAKER ANDG. N. WATSON,
p. 142.
(4] CHARLES JORDAN, Calculus of Finite Differences, Budapest, 1937.
[11 G.
BOHLMANN,2
'TbLewriter is indebted to Dr. Felix Bernstein for the reference to Bohlman.