TECHNICAL NOTE
ORP/CSD-76-1
A STATISTICAL ANALYSIS OF THE
PROJECTED PERFORMANCE OF
MULTI-UNIT REACTOR SITES
AUGUST 1976
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RADIATION PROGRAMS
WASHINGTON, D.C. 20460
-------
Technical Note
ORP/CSD-76-1
A STATISTICAL ANALYSIS OF THE PROJECTED
PERFORMANCE OF MULTI-UNIT
REACTOR SITES
Byron M. Bunger
Mary K. Barrick
August 1976
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RADIATION PROGRAMS
WASHINGTON, D. C. 20460
-------
PREFACE
The emphasis when performing this research was the
application of the formal procedures of statistics to the
multiple unit reactor siting problem. Expertise in the
technology of light water reactors was necessary to the
correct application of the statistics used here. James M.
Gruhlke and James W. Phillips of the Technical Assessment
Division very helpfully provided this expertise. They
also devoted a great amount of time and effort to the
collection and assessment of the data used here and to
the criticism of earlier drafts of this report.
-------
TABLE OF CONTENTS
Paqe
SECTION 1 INTRODUCTION AND SUMMARY 1-2
SECTION 2 EVALUATION OF DATA 3-13
Data for Pressurized Water Reactors 3-9
Sample Estimates of Mean and Variance for PWRs 9
Data for Boiling Water Reactors 9-11
Sample Estimates of Mean and Variance for BWRs 11-13
SECTION 3 METHOD OF ANALYSIS 14-19
Analytical Solution Based on Randomly Selected Monthly
Observations 14
The Transformation to Exposure Levels 15
Reactor "Down Time" 15-18
The Basic Premise Underlying the Methodology 18-19
SECTION 4 EXAMPLE SOLUTION AND RESULTS 20-31
Estimation of the Parameters of the Lognormal Distribution 20
Development of an Expression for the Percentile Point for
the Lognormal Distribution 21
Solution for PWR Assuming Ten Operating Months are
Representative of Annual Operation 21-27
Solution for PWR Assuming Twelve Operating Months are
Representative of Annual Operation 27-30
Results of Analysis of BWR 31
-------
FIGURE
Figure 1 PWR Primary Coolant 1-131 Concentrations for Individual
Fuel Loadings
TABLES
Paqe
Table 1 Dose Levels That Will Be Satisfied 95% of The Time
Table 2 PWR Primary Coolant 1-131 Concentrations
Table 3 BWR Noble Gas Releases Adjusted for Gland Seal and 30
Minute Delay
Table 4 Estimated Exposure Levels for PWRs
Table 5 Estimated Exposure Levels for BWRs
2
4-6
12-13
30
31
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SECTION 1
INTRODUCTION AND SUMMARY
The objective of this investigation is the statistical analysis
of the additivity of doses from multiple reactor sites. The problem
is hypothetical, but is of interest because it bears directly on the
question of how many reactors can be located at a single site under
provisions of the Environmental Protection Agency's proposed uranium
fuel cycle standard.1,2 The specific problem is to estimate the
potential dose that would result from a multi-reactor site where each
reactor individually just meets the provisions of Appendix I to 10
CFR Part 50.3
A basic premise underlying this analysis is that the maximum
possible dose to hypothetical receptor under the provisions of
Appendix I is allowed to exceed the 5 mrem per year design objective
whole body dose only under temporary and unusual circumstances.3 It
is assumed that one may assign a reasonable probability that this
dose is exceeded for any single reactor. For example, it is assumed
that the 0.25% fuel failure assumption commonly used, until recently,
as a design basis for pressurized water reactors is not exceeded, on
the basis of current operating history, more than 5% of the time.*
The implication of this assumption is that current operating data is
representitive of reactor operation that would exceed the Appendix I
limit of 5 mrem per year no more than 5% of the time. A conservative
approach is used for the purpose of this analysis; it is assumed that
the effluent control system is designed so that 5 mrem per year is
exceeded exactly 5% of the time.
Additionally it is assumed that when two or more reactors occupy
a site, each reactor operates independently. This means that the
operation or non-operation of individual reactors and the resultant
dose is independent of (i.e., is not correlated with) the operation
of other reactors on the site. A worst case set of assumptions is
used for modeling the multiple reactor cases in order that the
resultant doses be conservatively estimated. These assumptions are
that all reactors on a site occupy the same point rather than being
areally dispersed over the site, that the size of the site does not
increase as the number of reactors occupying it increases, that no
economies in received dose are to be realized from shared control
measures, and that those reactors exhibiting best performance are not
operated in preference for the other reactors at the site.
The data used in the analysis is taken from operating reports of
single unit reactors which are in commercial operation. Separate
analyses have been done for the pressurized water reactors (PWRs) and
the boiling water reactors (BWRs). The data for the PWRs is monthly
readings on the concentration of 1-131 in the primary coolant and for
BWFs is monthly readings on noble gas releases. Both sets of data
-------
are considered to be lognormally distributed. Section 2 contains an
evaluation of the data and a discussion of the assumption of
lognormality.
In order to use monthly emissions data from single unit reactors
in an analysis of annual doses from multiple reactor sites, a method
of transformation and statistical aggregation has been developed and
is described in Section 3. Briefly, it is assumed that monthly
readings on 1-131 concentrations (PWRs) and noble gas releases (BWRs)
are randomly selected from lognormal populations of monthly readings
whose parameters are estimated by statistical procedures. Monthly
readings are averaged to determine annual averages; then are assumed
to be linearly transformed to annual dose levels for single reactors.
These annual doses are then aggregated to determine the annual doses
from 4, 5 and 6 reactors on a site. An important part of the
solution is the premise concerning the distributions of annual doses
from single units and multi-units on a site. The premise is that the
true distributions are unknown, but that the Central Limit Theorem
gives assurance that the true distribution is somewhere between that
for the lognormal and normal distributions. Reactor down time has
also been treated in two ways, with the intention that calculated
estimates bracket the correct value.
An example solution is outlined in Section 4. This section also
contains the complete results of the analyses. These results are
summarized here.
Table 1
DOSE LEVELS (MREM/YR) THAT WILL BE
STATISFIED 95% OF THE TIME
4 Units 5 Units 6 Units
PWR 14 17 20
BWR 15 18 21
-2-
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SECTION 2
EVALUATION OF DATA
Data for Pressurized Water Reactors
The data set provides monthly readings on the concentration of
1-131 in the primary coolant of seventeen pressurized water reactors
(PWRs). Data is taken from the semiannual operating reports for the
respective reactors. There have been over 500 months of combined
operation of these seventeen reactors. For a variety of reasons, the
observations for many of these months either are not available or are
of limited or no use to this analysis. No observations are available
for over 150 months. The first year's operation is excluded because
it is not considered to be representative of typical reactor
operation. Additional information is lost because, on occasion, the
average of the readings for several months is provided instead of the
monthly readings themselves (since averaging over several months
results in the loss of information on the variation of monthly
observations, these averages were not made a part of the data set
analyized). In addition, the data for Ginna was excluded because it
is not considered to be typical of the other reactors. The result is
that the data set used for this analysis represents the concentration
levels for 132 months of operation and 23 months of "down time", when
the reactors are not in operation. The entire data set is listed in
Table 2.
Considerable effort was devoted to the inspection of the data to
determine how it could be best analyzed. A question of particular
concern is whether the variation in successive monthly readings on
the same fuel loading may be highly correlated. Another concern is
the relative variance of observations within a fuel loading as
compared to the variation between different fuel loadings. Questions
of this sort can be treated by techniques of regression analysis and
analysis of variance, but data limitations prevented these
approaches. The reason is that there are only five reasonably
complete sets of monthly readings for fuel loadings, and four of
these are from the same reactor, Haddam Neck. This effectively
prevents any elaborate analysis of the separate fuel loadings. A
simplistic approach to displaying the information on separate fuel
loadings is provided in Figure 1, which shows the mean and variance
of each of the 22 separate fuel loadings. Observations for the first
year's operation for each reactor are excluded. The five fuel
loadings represented by only the averages of monthly readings are
shown on the right hand side of the figure. Variance of monthly
readings cannot be calculated for these five fuel loadings. The
other 17 fuel loadings are ordered on the number of observations and
plotted from left to right in the figure. Many of these fuel
loadings are represented by six or fewer observations. The most
-3-
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Table 1
PWR PRIMARY COOLANT 1-131 CONCENTRATIONS (nCi/1)
January February
October
November
December
Location
Haddam Neck
(7/24/67)
Year
1967
68
69
70
71
72
73
74
75
1969
70
71
72
73
74
H.B. Robinson 1970
(9/20/70) 71
72
73
74
Ginna
(11/9/69)
Point Beach
(11/2/70)
1970
71
72
73
74
2.47(3)
2.3(2)
3.78(3)
3.4(4)
0
5.5(3)
2.99(3)
4.36(3)
NR
NR
NR
NR
0
NR
NR
NR
9.68(3)
NR
->-
NR
->-
2.6(2)
1.0(2)
2.98(3)
4.53(4)
2.5(4)
6.55(3)
5.89(3)
3.45(3)
NR
NR
NR
NR
0
NR
NR
NR
6.07(3)
NR
1.40(5)
NR
1.05(5)
2.9(2)
1.3(2)
1.48(3)
2.95(4)
2.7(4)
5.50(3)
3.78(3)
4.69(3)
NR
NR
NR
NR
0
NR
NR
0
1.30(4)
NR
NR
j.
4.7(2)
1.8(2)
1.34(2)
1.9(4)
0
5.19(3)
4.30(3)
3.92(3)
1.08(6)
NR
NR
NR
1.2(4)
NR
NR
NR
2.48(4)
NR
NR
-t-
1.5(2)
1.1(4)
0
2.54(4)
0
0
6.45(3)
0
5.31(5)
NR
NR
NR
3.1(4)
NR
NR
NR
0
NR
NR
0
NR
1.8(2) *9.0(1)
2.9(2)
0
1.74(4)
0
0
3.93(3)
4.7(5)
NR
NR
NR
3.8(4)
NR
NR
NR
0
NR
-e
NR
3.36(5)
2.6(2)
4.65(3)
1.58(4)
6.29(3)
0
4.1(3)
9.9(5)
NR
NR
NR
4.0(4)
NR
NR
NR
1.02(4)
->
-V
-*
->-
NR
3.68(3)
2.6(2)
5.86(3)
1.65(4)
6.83(3)
0
2.69(4)
7.5(5)
NR
6.1(4)
NR
3.6(4)
NR
NR
NR
1.12(4)
2.16(5)
»
NR
3.9(2)
4.0(2)
1.38(4)
1.56(4)
6.24(3)
0
4.22(3)
NR
NR
2.6(4)
NR
5.3(4)
NR
* NR
NR
NR
3.08(3)
1.41(4)
«-
6.2(4)
3.26(5)
NR
5.1(2)
1.31(3)
5.6(4)
2.13(4)
6.99(3)
0
4.33(4)
NR
NR
8.2(4)
NR
6.3(4)
NR
NR
NR
NR
4.75(3)
0
3.9(2)
1.3(2)
1.54(3)
2.79(4)
1.75(4)
6.51(3)
0
4.72(3)
NR
*NR
NR
NR
NR
4.8(4)
NR
NR
NR
NR
1.04(4)
NR
*
0
5 .2X2)
2.7(2)
9.10(3)
8.5(4)
1.76(4)
7.24(3)
5.14(3)
4.03(3)
NR
NR
NR
NR
NR
8.3(4)
NR
NR
NR
NR
2.37(3)
NR
-f-
0
«-
-e-
-------
Table 2 (cont.)
PWR PRIMARY COOLANT 1-131 CONCENTRATIONS (nCi/1)
January February March April May June July August September October November December
Location
Year
Palisades
(5/24/71)
Point Beach 2
(5/30/72)
Surry 1
(7/1/72)
Maine Yankee
(10/23/72)
Surry 2
(3/7/73)
Oconee 1
(4/19/73)
Indian Point 2
(5/22/73)
Fort Calhoun
(8/5/73)
1971
72
73
74
1972
73
74
1972
73
74
75
1972
73
74
1973
74
75
1973
74
1973
74
1973
74
NR NR NR
-» 3.4(4)
000
NR NR NR
+ 1.10(4)
1.13(3) 1.54(3) 3.65(3)
0 0 7.0(2)
0 1.25(5) 7.06(4)
NR NR NR
NR NR NR
4.7(1)
3.95(3) 3.04(3) *1.57(3)
2.87(3) 6.02(3) 4.87(3)
NR NR NR
0 0 8.4(2)
NR NR NR
NR
0
NR
2.31(3)
3.17(3)
1.43(5)
3.37(2)
NR
5.16(2)
2.35(3)
6.72(3)
NR
*NR
2.8(4)
NR
NR
*NR
0
NR
*NR
3.74(3)
2.06(4)
1.11(5)
1.74(2)
NR
5.94(3)
0
0
NR
NR
1.7(0)
*1.5(4)
NR
NR
NR
«-
0
NR
NR
*
NR
0
0
NR
-»-
-»
NR
2.76(3)*3.90(3)
2.30(4)
8.78(4)
2.3(1)
NR
2.94(3)
0
0
NR
NR
5.5(0)
6.0(4)
NR
1.29(4)
NR
0
3.85(3)
6.31(3)
NR
8.2(4)
1.1(2)
6.9(4)
1.83(2)
NR
0
0
+
7.
NR
4.24(3)
1.97(4)
NR
0
3.99(3)
4.41(3)
NR
1.1(5)
6.2(2)
7.6(4)
NR
*3.65(3)
NR NR
2.9(3)
0
0
2.
6.73(3)
8(3)
7.77(1)
3.05(3)
3.68(4)
NR
0
7.48(3)
4.19(3)
NR
5.75(4)
6.8(3)
5.1(4)
NR
5.71(2)
0
0
33(2)
-t-
0
1.05(3)
7.13(3)
NR
*NR
4.96(2)
2.16(3)
0
NR
8.2(4)
2.0(3)
5.4(4)
NR
3.41(2)
NR
0
0
0
0
2.53(3)
0
NR
NR
1.27(3)
3.86(3)
0
NR
0
0
5.4(4)
NR
3.20(3)
NR
0
0
-e
-<-
3.4(3)
6.75(2)
0
0
NR
NR
2.82(5)
4.91(3)
0
NR
0
0
5.1(4)
NR
1.88(2)
-------
Table 2 (cont.)
PWR PRIMARY COOLANT 1-131 CONCENTRATION (nCi/1)
January February March April May June July August September October November December
Location
Year
Oconee 2
(11/11/73)
Kewaunee
(3/7/74)
1973
74
1974
75
NR NR NR NR :
NR <1.0(1)
9.18(3) 2.97(4) *3.30(4) 2.16(4)
Three Mile 1974
Island(6/5/1974)
Oconee 3
(9/5/74
Rancho Seco
(9/16/74)
i
1973
1974
i
NR
NR
1.7(3) 7.1(2)
3.1(1) 7.3(1) 3.46(2) 4.17(4)
2.5(0)
9.2(3)
9.92(4)
6.4(3)
5.32(3)
NR = Not Reported
MDA = Minmum Detectable Activity
0 = Reactor not in Operation - Down Time
Numbers in parentheses represent exponents, for example: 3.9(2) = 3.9 x 102
* = Start of Second Year of Commercial Operation
-»<-= Average Concentration for Indicated Period
1.54(4)
<1.0(1)
-------
Figure 1
PWR PRIMARY COOLANT 1-13 1 CONCENTRATIONS FOR INDIVIDUAL FUEL LOADINGS
350
300
250
200
150
2 100
50
13 14 15 16 17
H H H SI
No. of
Obs. 22 20 17 13
Cum. No.
of Obs. 22 42 59 72
H S2
10
9 9
M 02 B2 P Bl Bl Bl
16 12 11 10 7
82 91 100 109 115 121 126 130 134 138 141 144 146 16 28 39 49 56
Individual Fuel Loadings
-7-
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conspicuous characteristic of the data apparent from this figure is
that the fuel loadings with the most information have the smallest
mean and variance. No completely satisfactory reason for this is
readily apparent. It might be proposed that the reactors with the
longest history have had time to work the "bugs" out but, this
explanation is not borne out by the data. The early history of the
reactors with the longest records do not show behavior similar to
that seen in the reactors with relatively short histories.
Differences in size of reactors does not appear to be an explanation.
A characteristic of the data that is important, but difficult to
evaluate, is that Haddam Neck has the longest history of any of the
reactors. Most of its data is good, so that its behavior threatens
to dominate the analysis. Four of the five longest fuel loading
records are from Haddam Neck.
The mean and variance of groupings of three, six and twelve
consecutive monthly observations were calculated. The variance
declined with increases in group size, but data was lost with each
increase in group size, because some reactor histories had less than
three or six observations. Therefore, the results are difficult to
interpret.
The tentative conclusion drawn from the results of the grouping
on three, six and twelve months as well as from the grouping on
separate fuel loadings is that individual fuel loadings probably
exhibit different variances. This proposition could be more fully
explored if there were more complete records for the newer reactors.
As things now stand, however, there is insufficient information to
proceed with the analyses of separate fuel loadings.
Another approach is to consider each observation to be a member
of a "grand" population of monthly observations. Although this
approach may be overly simplified, it is tractable. It is the
approach used in this analysis. The implications of the assumptions
underlying this approach will be discussed in Section 3.
One necessary step when using this approach to the analysis is
the determination of the form of distribution of the observations.
Professional opinion is that the monthly observations are
lognormally distributed. Therefore the data was tested for normality
and lognormality by chi-square goodness of fit. The 132 observations
representing operating date were tested and found to fail the test
for normality and lognormality at the 5% level of significance.
However the data was found to more closely fit the lognormal than the
normal distribution.
-8-
-------
The operating data for Haddam Neck was then tested separately
for lognormality. The chi-square goodness of fit test verified that
the concentration levels for Haddam Neck are adequately described by
a lognormal distribution (test not significant at the 5% level).
This provides some evidence that the reactors with the longer
operating experience do display a lognormal distribution.
Analysis was continued on the assumption that the whole data set
of 132 observations is lognormally distributed. The justification
is: 1) professional opinion that monthly concentrations are
lognormally distributed and 2) the test showing that the Haddam Neck
data is lognormally distributed. Haddam Neck is also the best data
available. The assumption that the distribution is lognormal implies
the expectation that a more complete data set for the other reactors
would determine a lognormal distribution for the entire set of data.
Sample Estimates of Mean and Variance for PWRs
The analysis to be performed here will require that the data be
expressed in its original form and also in its log form. The mean
and standard deviation for the 132 months of operating data, in
original form, expressed in nanocuries (10~9 Ci) per liter are:
* = 2.108 x 10*
.sx = 3.602 x 10* (1)
The mean and standard deviation of the 132 months of operating data
expressed in log form are:
9 = 8.778
sy = 1.772
where g -, £ £ y. - £ £ / x. anc/ s -. ^ Z(^ y)* = ~~ Z (i» *,-j
sy = 1.772 (2)
P§ta for Boiling Water Reactors
Monthly releases of noble gases from single-unit Boiling Water
Reactors (BWRs) form the basis for this analysis. The release data
is taken from the semiannual operating reports for each reactor. The
noble gases being considered are those normally reported by the
utilities; namely Kr-85m, Kr-87, Kr-88r Xe-133, Xe-135 and Xe-138.
Information has been obtained for the 6 reactors with at least 12
months of commercial operating experience as of December 31, 197U.
The first 12 months operation have been discarded as atypical of
future performance. The data set thus defined comprises 18U months
of operating experience including 27,months of "down time." As with
the previous analysis, these monthly releases are assumed to be
-------
representative of random monthly releases from a typical single-unit
reactor currently in operation.
Although the data (stack releases reported by the utilities) may
be assumed to represent current operations, there are at least two
reasons why they are not as yet in the proper form for this analysis.
Therefore the data requires further modifications.
The first problem involves the off-gas treatment system. The
present data come from reactors with various systems for treating
releases from the steam jet air ejector. In modeling releases the
normal procedure is to use a source term based on a 30-minute hold-up
system regardless of the system actually in use at a particular
reactor. Therefore the data should be recalculated using only a
30-minute delay system.
A difficulty encountered in using the above modification is
determining the fraction of a reported stack release which has passed
through the hold-up system and therefore should be adjusted. To do
this, the estimated gland seal system release for each isotope (taken
from AEC figures4) has been subtracted from the stack release for
that isotope and the remainder, if any, has been recalculated based
on 30 minutes delay. The sum of the adjusted releases and the gland
seal releases for all six isotopes constitutes the estimated stack
release adjusted to 30 minutes delay.
The second problem involves the size and operating level of
future reactors. These facilities are assumed to have a generating
capacity of 1000 MWe (3UOO MW(th)) and to operate on the average at
80% of that capacity. Current reactors are smaller and on most
occasions operate at a lower percent capacity. It is also known that
noble gas release levels are related to power generation. In order
to use the available data, a linear relationship between releases and
power generation has been assumed. With such an assumption a monthly
release can be scaled easily to the equivalent release expected from
any given size reactor for any specified percent capacity.
To scale the data, a monthly release (normalized to 30 minute
hold-up) is first divided by the power generated in that month and
then multiplied by constants to reflect a 3400 MW (th) plant
operating at 80ft capacity. Since the constants apply to all the data
and therfore will have no bearing on the underlying distribution,
they have been omitted from the scaling process. The releases will
thus be in units of Curies /MWd(th). The major factor in the
analysis is the relationship between the mean and variance of the
underlying distribution. This relationship will be the same with or
without the constant multiplier. It should be noted, however, that
by not employing the constant multiplier, the transformed data will
-10-
-------
not reflect the expected absolute amount of releases and therefore
should not be used in other analyses where such values are required.
The modified data set used in the analysis is shown in Table 3.
With the data set redefined so that it fits the problem to be
investigated here, the first step in the analysis is to determine if
the 157 data points representing actual operation are lognormally
distributed. The precedure involved is a chi-quare goodness of fit
test. Using this test, it can be shown that the operating data fit a
lognormal distribution (test not significant at the 5% level).
Sample Estimates of Mean and Variance for BWRs
The mean and standard deviation for the 157 months of operating
data, in the original form, expressed in curies per megawatt day
thermal are:
x = 2.196
5X = 2.647 (3)
The mean and standard deviation for the 157 months of operating data,
transfered to log form, are:
3 = 0.140
Sy = 1.268 (4)
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Table 3
BWR NOBLE GAS RELEASES ADJUSTED FOR GLAND SEAL AND 30 Min. DELAY (Ci/MWd(th))
January
February March
June
July
Location
Oyster Creek
Nine Mile Point
Millstone
Year
1969
70
71
72
73
74
1969
70
71
72
73
74
1970
71
72
73
74
-j-
->-
4.52
2.89
1.88
->
6.02(-2)
8.55C-1)
3.03
->
8. 17 (-2)
5.36C-1)
0
6.49(-l)
4.55
6.07
2.19
5.69C-3)
1.16(-1)
9.40(-1)
3.24
1.61
1.82C-1)
3.83(-l)
0
3.9K-1)
6.21(-1)
1.65
5.45
1.02(1)
2.74
-<-
5.27(-l)
1.85
5.86
-4-
1.82(-1)
6.56(-l)
1.93C-1)
5.50(-1)
7.19
1.02(1)
2.92
-*-
2.83
3.28
7.34
->
1.64(-1)
4.86(-l)
4.70(-1)
5.15C-1)
9.15
0
0
1.5K-1)
0
0
0
4.30(-1)
1.40
0
8.27(-l)
-<-
-(-
1.21
1.20
0
4.03(-1)
2.15(-1)
4.33(-l)
-Ir
3.74(-l)
3.97
0
7.29(-l)
-?-
1.84
9.57(-l)
1.88
9.43(-l)
4.42(-l)
7.75C-1)
6.59(-l)
->-
2.74(-l)
3.53
7.59(-2)
2.73
3.27
9.75(-l)
2.14
9.3K-1)
5. 76 (-2)
5.56(-l)
1.39
6.94(-l)
2.05
4.96(-l)
5.40
2.02(-1)
8.30
->-
k
3.63
1.23
1.78
7.82(-l)
1.98
9.87C-1)
6.96(-l)
&-
1.20
0
3,52(-l)
0
7.37C-1)
0
1.48
1.98
1.39
5.95(-l)
5.80(-1)
8.86(-l)
-J-
1.45
0
3.79C-1)
0
3.81(-1)
1.18
2.07
1.54
1.95
-«-5.63(-3)
*
1.33
2.01
1.44
1.82
-»-3.41(-2)
*1.14
0
3.46(-l)
1.98
-«-
-
-------
Table 3 (cont.)
BWR NOBLE GAS RELEASES ADJUSTED FOR GLAND SEAL AND 30 Min. DELAY (Ci/MWd(th))
January February March
April
May
June
July
August
September October November December
Location Year
Pilgrim I 1972
73 2.45(-l) 2.12(-1) 2.94(-l) 4.80(-1)
74 0 0 0 0
Monticello 1971 ->- 3. 18 (-2)
72 4.71(2) 5.69(-l) *8.17(-1) 7.15(-1)
, 73 2.53 1.93 0 0
i- 74 7.89 8.27 7.29 0
I
Vermont Yankee 1972
73 5.56 0 6.40(-1) 7.17(-1)
74 1.85(-2) 2.3K-1) 1.30(-1) 3.52(-l)
7.05(-3)
5.27(-l) 3.58(-l) *3.38(-l)
0 0 3.17(-1)
3.78(-2)
1.35 1.16 1.19
3.29(-l) 5.27(-l) 8.03C-1)
3.23 4.37 6.47
6.80(-1) 5.47(-l) 4.08(-1)
2.9K-1) 7.99(-2) 3.94(-l)
0
5.06C-1)
4.43(-l)
4.39(-l)
2.34
1.25
6.05
4.50(-1)
8.96(-2)
1.12(-2)
1.22
2.12
5.0K-1)
2.68
1.38
2.02(1)
7.25(-3)
*9.85(-l)
1.92C-2)
3.66C-1)
2.99
4.75C-1)
2.37
2.51
7.19
2.84(-l)
0
7.23C-1)
1.56C-1)
2.59(-l)
6.51
5.47(-l)
3.36
5.20
2.44
8.63C-1)
1.60(-1)
0
2.38(-l)
2.23(-l)
1.66
0
3.18
4.57
8.40(-X>
1.62
2. 31 (-2)
4. 74 (-2)
0 = Reactor Not in Operation - Down Time
Numbers in parentheses represent exponents, for example: 3.81(-1) = 3.81 x 10
* = Start of Second Year of Commercial Operation
-»-<-= Average Release for Indicated Period
-------
SECTION 3
METHOD OF ANALYSIS
Analytical Solution Based on Randomly Selected Monthly Observations
The objective of this analysis is to estimate the exposure from
a multi-reactor site under the restriction that each reactor, if
tested individually, would just meet the Appendix I operating
criteria (five millirem per year). Since this limit and the uranium
fuel cycle standard are both expressed in millirem per year, the
distribution of exposures must also be expressed in annual terms
(i.e., millirem per year).
Based on the assumption that the 132 observations on 1-131
concentrations and the 157 observations on noble gas releases are
lognormally distributed, there are at least two conceivable ways to
proceed to an estimate of the mean and variance of the annual
exposures. The first is to use simulation techniques to determine
the distribution of the annual mean concentration, then transform to
annual exposure and to again simulate to determine the distribution
of the mean level of exposure from multiple reactors on a site. This
method is not pursued in this analysis. The second possible method,
the one adopted here, is to use an analytical technique for
estimating as closely as possible the distribution of the mean annual
concentration levels and, after transformation to exposures, to again
use an analytical technique to approximate the distribution of the
exposures from a number of reactors on a site. The technique for
performing this analysis is discussed in detail later in this
section.
The method used here, as well as the simulation method discussed
above, relies on a concept that needs special attention. Both
methods assume that the annual average concentration (PWRs) and noble
gas released (BWRs) are represented by the mean of individual monthly
observations selected randomly from their respective populations,
independently of reactor and fuel loadings. Since a number of
different fuel loadings are represented in the data, this assumption
is quite different from that underlying an analysis based on the
separate fuel loadings, as discussed in Section 2.
Aside from the argument that the method used here is justified
because it is manageable, it can also be argued that when the
analysis is complete, with a model for four or more reactors on a
site, the annual mean emissions level is based on the sum of 48 or
more monthly observations from at least four fuel loadings. At this
level of aggregation, it may make little difference whether the 48
individual observations were randomly selected or not, so long as
they are representative of typical reactor behavior.
-14-
-------
The Transformation to Exposure Levels
The Appendix I limitation and the proposed uranium fuel cycle
standard are expressed in terms of radiation exposures. Therefore
this analysis must assume some form of transformation from coolant
concentrations (PWRs) and stack releases (BWRs) to exposure levels.
Linear transformations will be assumed. For example, the
distribution of radiation emissions from PWRs that cause exposures
will have the same distribution as that assumed for the concentration
levels (i.e. if the coolant concentration is assumed to be
lognormally distributed, radiation exposures at a given location
will also be lognormally distributed). The linear transformation
will also transform zero observations for coolant concentration to
zero levels of radiation exposures.
Reactor "Down Time"
Reactor "down time" is a problem that must be given special
attention, when shut down for refueling or repairs, past practice
has been that 1-131 concentrations in the primary coolant for PWRs
have not been reported. Stack releases for BWRs are not measured
during shutdown. Therefore readings of zero have been assumed for
months when the reactors are out of operation. Since there are
numerous months of down time in the data, and since it is to be
expected that even the best future experience with reactors will
still include frequent down-time periods (if for nothing more than
refueling), the zero level readings should be included in the data
set. The "zero" readings account for about 1/6 of each of the data
sets. They occur frequently enough that they cannot be included in
the data set for operating months, since the lognormal distribution
does not provide for any observations at zero and provides for only a
limited number at values close to zero. Therefore operating months
must be handled seperately from down time. The following is a
discussion of the two possible approaches to the treatment of these
zero values.
The months when there is no data representing coolant
concentration levels (PWRs), because of refueling or repairs, may be
handled in at least two ways. The first is to assume that a zero
level of exposure (after transformation from a zero level
concentration in the primary coolant in PWRs, for example) is a valid
representation of the true exposure level. This, of course, implies
that the major source of radiation exposure is leakage from primary
coolant and little is contributed by such things as wastes stored at
the reactor site, or from other plant operations releasing
radioactivity more or less independently of power production, such as
containment pipe purge or waste gas decay tank releases. The second
method is to assume that the emissions level remains relatively high
during periods of refueling and repair, so that exposures are best
-15-
-------
represented by assuming that the monthly concentration levels, after
transformation to exposures, stay at the same levels. This method
would assume that exposures are just as high during periods when the
reactor is down as when it is operating.
It is probable that neither of these methods is completely
satisfactory. The first may underestimate the true exposures,
especially for PWRs. The second may overestimate exposures for the
BWRs, but may closely approximate them for the PWRs because it has
been observed that radioactivity releases for PWRs remain quite high
during shutdown periods, especially noble gas releases. Both methods
will be used in this analysis, to see what impacts the different
methodologies have. The first analysis will be performed assuming
that exposures are zero when reactors are down for refueling or
repairs. Later it will be assumed that down time is represented by
the same exposure level as operating time. Each year is, therefore,
assumed to be represented by twelve months of operating data. Each
of these techniques will be more fully elaborated upon in Section 4,
where, the calculations are discussed. With appropriate
modifications, similar arguments apply to BWRs.
The data on which the analysis of the PWR is based has 132
months of operating data and 23 months of time of no operation
because of refueling or repairs. The total months of operation are
132 + 23= 155. The 23 months of down time represent 23/155 = IU.8%
of the time, which is close to 1/6 of the time, or 2 months out of
each year. Therefore, it will first be assumed that there are 10
months of operation each year and 2 months of down time.
It will be assumed that the observations on operating months and
on "down time" represent independent distributions, and that
independent monthly observations are drawn from each of them. For
each year's data set, ten observations are drawn from the operating
data and two from the "down time" data. It is assumed that this
holds true for every year's operation. This is, of course, a
restrictive assumption, but facilitates the solution. To combine the
two distributions into one creates a population that is not easily
treated, for simulation techniques would be needed to estimate the
distribution of a sum of observations drawn from this distribution.
Distribution of "down time" observations
(weight = 2/12)
Distribution of operating observations
(weight = 10/12)
Concentration
-16-
-------
The areas represented on the sketch above should be in
proportion to the weight given them in drawing samples. The
distribution of "down time" is a narrow vertical strip approaching
zero width, but whose area is 1/6 that of the distribution of
operating months. The variance of these "down time" observations is
equal to zero.
Since the "down time" observations have zero variance and since
the proportion of sample observations from the two distributions is
constant, the variation in the operating data accounts for the entire
variation observed in the yearly performance. Yearly performance is
characterized by the weighted average of the mean of ten observations
drawn from the distribution of operating data and the mean of two
zero observations from the distribution of "down time". The
following sketch represents the distributions of these two means.
Distribution of mean of two observations
on "down time".
-Distribution of mean of ten observations
on operating data.
Concentration
Let 8V- determine a tail area that represents 5% of the total
area for the distribution on the mean of ten months of operation.
Averages over ten months of operation that exceed £^~ can be expected
5% of the time. Since the only source of variation in a years
performance is in the operating data, the mean value for a years
operation will be directly proportional to the average for the ten
operating months in each year. Therefore, the tail area represents
the probability (5%) that the annual average concentration will
exceed (10/12) £fj . The (10/12) £w represents the weighted
average of ten months of operation and two months of zero readings
observed when the reactor is not in operation.
For the second case to be investigated, it will be assumed that
reactor down time contributes as much to exposure as does operating
-17-
-------
time. This case is handled by assuming that all 12 monthly
observations are drawn from the distribution of operating data.
Therefore, no weighting of operating and down time months is
necessary-
The analysis of the BWR case is similar. The data set
represents 157 months of operation and 28 months of down time for
refueling and repairs. Down time represents 15.255 of the total
months of operation (27/185), approximately 1/6 of the time.
Therefore, the same two cases used to investigate the PWRs are used
for the BWRs, one assuming two months down time with emissions equal
zero and the other assuming the two months down time having emission
levels equal to those experienced when in operation.
The Basic Premise Underlying the Methodology
This analysis is based upon the use of the Central Limit
Theorem, one of the fundamental theorems of statistics. Loosely
expressed, this theorem states that the distribution of the sum of
randomly selected observations from any population will approach
normal as the number of summed observations increases. The great
strength of this theorem is that there are no restrictions on the
distribution from which the observations are selected, except that it
have a finite variance. Based on this theorem, the sample average,
which is a sum of randomly selected observations divided by the
sample size, will approach a normal distribution as the sample size
is increased. It is generally accepted that a sample size greater
than thirty is sufficiently large to assume that the sample average
is normally distributed.5
The data for both types of reactors (PWRs and BWRs) are monthly,
and the standard against which exposures (after transformation) are
to be compared is expressed in annual terms. The Central Limit
Theorem cannot be exercised to claim that the annual average
concentration is normally distributed, because the annual average is
based on only twelve monthly readings. However, the Central Limit
Theorem is still of use, for it is necessary to the method of
analysis developed here. This methodology will now be described.
A sample of size one selected from a parent population will have
the same distribution as the parent distribution. In most cases the
mean of samples sized two and larger will take a form different from
that of the parent population, and this difference will increase as
the sample size increases. According to the Central Limit Theorem,
the distribution of the mean will approach a normal as the sample
size increases.6 The basic premise upon which this solution is based
is that the distribution of the mean of a sample of 10 to 12
observations will describe a distribution that is somewhere between
that of the parent population and the normal. Even though the
-18-
-------
precise distribution the mean will follow is unknown, it must be
somewhere between that of the parent distribution and that of the
normal.
Specifically, the area in the right tail of the distribution of
the mean is of interest. The probability that the true value of the
mean will exceed some specific value in the right tail decreases as
the sample size increases. For any selected sample size, this
probability will be somewhere between that which would be measured if
the sample mean were distributed as the parent distribution and that
which would be measured if it were normally distributed.
The problem of interest here is not the determination of the
area enclosed in the right hand tail, but is, instead, the closely
related problem of determining the value that would enclose an area
in the right tail that is equal to a selected percentage of the area
under the entire distribution (e.g. 5%). The premise upon which this
solution is based can, therefore, be restated: it is that the value
that cuts off 5% of the area under the right tail of the true
distribution of the sample mean lies somewhere between the values
that would enclose 5% tails of the parent distribution and the normal
distribution, where the parameters specifying the parent and normal
distributions are based on estimators calculated from the sample
data. In order o apply this methodology, the distribution of the
parent population must be known or assumed. As discussed in Section
2, the parent population is assumed to be lognormally distributed.
-19-
-------
Section H
EXAMPLE SOLUTIONS AND RESULTS
Estimation of the Parameters of the Lognormal Distribution
The methodology used here requires estimates of the mean/w and
variance^2 of the lognormal distribution of concentration levels
(for PWRs) and noble gas releases (for BWRs). The properties of
different estimation techniques are discussed in Chapter 5 of
Aitchison and Brown.7 A variety of techniques are available. Each
can be expected to provide slightly different estimates of the
parameters/^ and
-------
Development of an Expression for the Percentile Point for the
Lognormal Distribution
Before working through an example solution, it will be useful to
develop the expression for the value of Lv, which cuts off an area in
the right hand tail of the lognormal density equal to (1-v). From
Aitchesin and Brown, p. 9, the expression for L^ is:
L f . . , t
O y Jo \ AA T" / CJ J
v ** P V" z" '
where z.v is the value that cuts off an area in the right hand tail of
the standard normal density equal to (1-v). This expression is
needed because no table of areas under the lognormal density are
available, so corresponding areas under the normal density are used.
In order to use this formulation for Lvr the values^ and o- are
required. Since the methodology developed here is not directed
towards the estimation of /J and
-------
lognormal distribution were given in equations (5). Let gm and hm
represent estimates of the sample mean and standard deviation
respectively. Therefore:
gm - a - 3.027*. \o*~
hm = b = (1,555* to'*)* - 1.2*1 x lo* (8)
These are estimates of the parameters of the distribution of
monthly operation, from which a sample of 10 months operating data
are drawn. The sample of 10 monthly concentration levels will have a
mean q and standard deviation hA .
^ = dm = 3.0Z7 ^ 10 +
= 3,1 56* 1O* (9)
These paramenters can be used to determine the value of
relying on the derivation developed in equation (7).
i , r/3
Where: w - In (9%a + // = /" [ ( 3
= 3.
The transformation to exposures can now be performed. In
performing this transformation, no attempt is made to develop a
pathway model. Instead, it is assumed that the concentration level
represented by £yj. corresponds to the operating criteria established
in Appendix Ir which determines a 5 millirem annual limit for each
reactor. The correspondence between £9J. and 5 millirem per year is
assumed to be represented by a linear transformation. In making this
transformation, it has been assumed that the 95% point for
concentration (£<,?) can be linearly transformed to the 95% point for
exposure. Equating the 95% point for exposure to 5 mrem imposes an
interpretation on the Appendix I limit; that the 5 mrem limit is not
to be exceeded more than 5% of the time under normal operation. A
worst case has been assumed in this solution; that 5 mrem is exceeded
exactly 5% of the time. This same linear transformation can be used
to transform the mean annual concentration to mean annual exposure
and also the standard deviation of annual concentrations to that for
-22-
-------
annual exposures. There is, however, one conceptual clarification
necessary before performing the transformation. The primary interest
is the transformation of the paramenters of the distribution of
operating data, since it is reasonable that a zero concentration
level for "down time" will transform to zero exposure. Since the
operating data represents only ten months operation, the
concentration level will be assumed to transform to exposure levels
that produces 5 millirem during a whole years operation, rather than
5 millirems for ten months operation. Therefore the parameters of
the distribution for operating data is weighted by the factor 10/12.
The following calculations determine the transformation factor.
to/ c -r- _ c-
'12, (-^j / J tn re./*
r = */(%£)
The mean
-------
It should be noted that these equations do not necessairly imply
that all the r reactors operate the same ten months out of each year.
The down time for different reactors could be spaced at different
times throughout the year.
Either of two assumptions can be made concerning the
distribution of the exposures from the aggregation. One assumption
is that the aggregation continues to be lognormally distributed and
the other is that it is normally distributed. These two alternatives
come from the basic premise upon which this solution is based, that
these two distributions will "bracket" the true distribution, as was
discussed above in the section on methodology.
First, it will be assumed that the aggregation is lognormally
distributed. The value of r will be set to equal six for
demonstrating the technique.
= 6 (I, 9/3)
= // f Id
(2. f 18)
= 6.111 (10)
The exposure level that "cuts off" a 5% tail is determined by
the formula introduced earlier, in equation (7) .
J-T- / r/ A in \* , /7
In this case: w = //? i[}j~?re/ + (j
Therefore: i^. = 23.
These parameters represent the distribution of exposures from
six reactors, for ten months of operations. The exposure level for
the other two months operation for each reactor is equal zero, so,
taking the weighted average to determine the 95% level will give:
/£
"W ~ 12
= /?.
-24-
-------
This completes the demonstration for the case assuming annual
concentration levels and annual exposure levels are both lognormally
distributed. It will now be assumed that annual concentration levels
continue to be lognormally distributed, but that the exposure level
for the aggregation of six reactors will be normally distributed.
The same mean and standard deviation estimates calculated in equation
(10) still apply, but the equation for i)9f will now be:
= 21.
Therefore: Rq5 ~ Ti. 2/. ^Tf - II. 153
This completes the analysis based on the assumption that the
annual concentration is lognormally distributed. It will now be
assumed that annual concentration is normally distributed as was
discussed in Section 3. The sample estimates based on the data in
its original form serve as the starting point. Let the mean and
standard deviation of the distribution of the annual average under
this assumption be y' and h'A . From equations (1) :
a' - d = 2,108 X 10*
JA
h' - b/VTo = (3. 60-2. XiO* )///#
f\
- US'? x io4
The 95th percentile point for the normal distribution is represented
by e,'f .
= 2.1^8 x io*+ t'tff (i 139 x io4,
"" ^y & Cf o V" / /*)
-" "1 i tj ^ *\ I u
VX ' «p^ *^
The transformation to exposures will give:
ff £*r 7- = 5
r - i-507 x la'*
-25-
-------
The mean g and standard deviation h£ of the ten month exposure level
is found to be:
a' ~ To' ~ I.5O7 xio~* (2.108 xic*)
JE JA
- 3.177
/),' - r/,; = i. so7 *t*~+ (MSI*!**)
£" rl
= \.116
Note again that these parameters represent the distribution for
ten months of operation only. They must be weighted by the two
months of down time to attain the annual values.
Aggregation to 6 reactors on a site gives:
a' - 6 a' = 6(3.177)
J 6 + £
h6 ~ ~6 V - -6 (1.716)
= 4.203
Since the distribution of the mean for ten months operation has
been assumed to be normal, the aggregation to six reactors will also
be normal .
-0'^ - 11.062 + 1.6+5 (4,
- 25.
-------
Method 1 (Lognormal at one reactor, lognormal at
six reactors) : 19.2 mr em/year
Method 2 (Lognormal at one reactor, normal at
six reactors): 18.0 mrem/year
Method 3 (Normal at one reactor, normal at
six reactors: 21.6 mrem/year
Solution for PWR Assuming Twelve Operating Months are Representative
of Annual Operation
As discussed above, it might be argued that the exposure level
observed from a reactor is best assumed to be the same when down for
repairs as when in operation. Therefore, it will now be assumed that
twelve monthly observations are representative of a years operation.
The solution is similar to that used when assuming ten operating
months represents annual operation. Using equations (8) :
= 3.029 x 10 +
- 1.251 XlorT2 = 3.6i\ x lo+ UD
A lognormal distribution will now be fitted to these parameters,
for the purpose of determining the 95% point (equation 7) .
73 JA
\V = 'fl
= O. 881
-} (a 88*) + I. 6+5 (0.
The transformation to exposures will give the transformation
factor (T) .
T = 5
c?a .5-
r -
The mean and standard deviation of the exposure level for one
year is:
5L = 5. 4- 6 7 *
= /- 657
-27-
-------
Aggregation to six reactors will give
q = 6(1. 657)
J6
h6 = -ie ( 1. 175)
= f 838
Assuming that the aggregated mean annual exposure level follows
a lognormal distribution, it is found that:
- 0.213
J (0.213) + I. e + 5 (0-2 13)* J
- 1*7. on
Since no adjustment for "down time" is necessary, the value for
is:
This completes the solution, assuming the aggregated mean for
six reactors is lognormally distributed. Now, assuming that the
aggregated annual exposure level follows a normal distribution, it is
found that:
. 838)
- 17.
This completes the analysis for the cases assuming the annual
concentration level is lognormally distributed. The case assuming
annual concentration to be normally distributed will now be
demonstrated. As for the ten months case, sample estimates of the
data in its original form must be used. From equations (1) :
<7 = # - 2. IO8 x /0*~
JA
*
= ( 3, 6 &2 X 10
~ / 04- £> X / £>*
f = 2.1 08 x \c* +
~ 3, &,
-------
Performing the transformation to exposures
£' T -- 5
c
-------
Method 2 (Lognormal at one reactor, normal
at six reactors): 17.9 mrem/year
Method 3 (Normal at one reactor, normal
at six reactors): 22.0 mrem/year
These results, as well as those for the analysis assuming zero
contribution to exposure when the reactors are not in operation, are
summarized in Table 4. This table also summarizes the results of the
analysis for aggregates of 4 and 5 reactors on a site. Solutions for
the cases of 4 and 5 reactors on a site have not been demonstrated
here, but are similar to those for the 6 reactors cases.
The averages of the six solutions investigated for the 4, 5 and
6 reactors on a site cases are shown. The summary results provided
in Section 1 are based on these averages. The two solutions assuming
lognormal distributions at the annual level and normal distributions
at the aggregated level (methods LN-N) appear to be the most
realistic, considering the implications of the Central Limit Therom.
Since the solutions based on this method have the lowest estimated
exposure levels, the use of the average of all methods of solution
for the estimate of the exposure levels is probably conservative.
Table 4
ESTIMATED EXPOSURE LEVELS FOR PWRs
Reactor Operation Number of Reactors on a Site
Method1 Months per year 4 5 6
LN-LN 10 14.22 16.8
LN-N 10 13.2 15.6
N-N 10 15.3 18.5
LN-LN 12 14.1 16.6
LN-N 12 13.1 15.5
N-N 12 15.5 18.8
Average 14.2 17.0 19.6
* Method refers to the distribution assumed in the solutions. LN
means lognormal and N means normal. They are in the order of their
incorporation into the solutions.
2 units are mrem/year
-30-
-------
Results of Analysis of BWR
Solutions for the BWR are completely analogous to those for the
PWR. The only difference is that the starting point for the BWR
analysis is stack release data rather than the coolant concentration
data used for the PWR. Sample estimates of mean and standard
deviation for the data in original form and after transformation were
provided in equations (3) and (4). Maximum likelihood estimates of^
and a-2 were given in equations (6). The stack releases are assumed
to be linearly transformed to exposure levels in a manner similar to
that for the PWR. After transformation to exposure levels, the
solutions are exactly the same. They will not be demonstrated.
Table 5 summarizes the results of the analysis of the BWRs. The
averages shown are summarized in Section 1, The discussion for Table
4 is appropriate for this table as well. As for the PWR cases, the
solutions assuming lognormal distributions at the annual level and
normal distributions at the aggregated level appear to be the most
realistic. Therefore, the use of averages for the BWR cases also
appears to be conservative.
Table 5
ESTIMATED EXPOSURE LEVELS FOR BWRs
Reactor Operation Number of Reactors on a Site
Method1 Months per year 4 5 6
LN-LN 10 14.52 17.u 20.2
LN-N 10 14.0 16.8 19.6
N-N 10 16.1 19.7 23.2
LN-LN 12 14.7 17.7 20.6
LN-N 12 14.2 17.2 20.1
N-N 12 16.4 20.0 23.5
Average 15.0 18.1 21.2
1 Method refers to the distribution assumed in the solutions. LN
means lognormal and N means normal. They are in the order of their
incorporations into the solution.
2 Units are mrem/year
-31-
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REFERENCES
1. Environmental Radiation Protection for Nuclear Power Operations:
Proposed Standards, Federal Register, HO:, No. 101, 23420-23425,
May 29. 1975.
2. Draft Environmental Statement: Environmental Radiation
Protection Requirements for Normal Operations of Activities in
the Uranium Fuel Cycle, U.S. Environmental Protection Agency,
Office of Radiation Programs, May, 1975.
3. Opinion of the Commission in the Matter of Rulemaking Hearing,
Numerical Guides for Design Objectives and Limiting Conditions
for Operation to Meet the Criterion "As Low as Practicable" for
Radioactive Material in Light-Water-Coolant Nuclear Power
Reactor Effluents, Docket No. RM-50-2, U.S. Nuclear Regulatory
Commission, May 5, 1975.
4. Numerical Guides for Design Objectives and Limiting Conditions
for Operation to Meet the Criterion "As Low As Practicable" for
Radioactive Material in Light-Water-cooled Nuclear Power Reactor
Effluents, United States Atomic Energy Commission, WASH-1258,
July 1973.
5. For a discussion of this point, see: Maurice G. Kendell and
Alan Stuart, The Advanced Theory of Statistics, (Hafner,
Publishing Company, New York 1958) Vol. I, pp.193-195 and
pp.223-224.
6. For example, see: Alexander M. Mood and Franklin A. Graybill,
Introduction to the Theory of Statistics, Second Edition (New
York, McGraw Hill Book Company, p.152, 1963).
7. J. Aitchison and J. A. C. Brown, The Lognormal Distribution
Cambridge, Cambridge University Press, 1957.
8. Finney, D. J., "On the Distribution of a Variate Whose Logurithm
is Normally Distributed," Journal of the Royal Statistical
Society, Supplement 7, 1941.
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