Citation: Papadopoulos A (2011) Stochastic Ontogenetic Allometry: The Statistical Dynamics of Relative Growth. PLoS ONE 6(9):
e25267.
https://doi.org/10.1371/journal.pone.0025267
Editor: Zheng Su, Genentech Inc., United States of America
Received: April 15, 2011; Accepted: August 31, 2011; Published: September 23, 2011
Copyright: © 2011 Anthony Papadopoulos. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: Support for this study was provided by National Science Foundation (NSF) award DEB-0616942 to Sean H. Rice. NSF had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The author has declared that no competing interests exist.
Introduction
The most notable contributor to the mathematical analysis of allometry is J. S. Huxley, who in 1924 published a seminal paper in which he proposed that the power-law function (
) be used to describe allometric growth [1]:
where y is the size of a trait, x is the size of another trait, and b and k are useful descriptors of allometric growth [1], [2]. Since then, numerous papers that support 
as a model for allometric growth have been published. One paper, in particular, shows that 
is an axiomatic functional form of allometry [3]. In theory, this suggests that the composite model 
, in which extrinsic time t is treated explicitly, yields an exact correspondence between 
and 
, assuming that there is no stochasticity in 
[4]:
where 
and 
are the sizes of two ontogenetically related traits at any given t [4]. In reality, however, 
and 
are inherently correlated stochastic processes, which are correlated random variables that depend on the deterministic variable t. It is not known with certainty the value of 
and the value of 
until after their measurements have taken place. Thus, 
is exact, but unrealistic, only as a deterministic model. Subsequently, when the relationship between the realizations of stochastic 
and the realizations of stochastic 
is described by 
, the probabilistic version of either b or k is not always constant with t. In fact, as this paper will show, the statistical moments of the random ontogenetic growth function for 
and for 
affect the temporal distribution of both b and k. This phenomenon has significant implications with regard to organismal form and function. And so the objectives of this study are to first incorporate stochasticity into 
by treating 
and 
as correlated stochastic processes, thereby formulating an exact probabilistic model for allometric growth that applies throughout the ontogeny of any organism, and then to analyze the ontogenetic statistical moments that specifically govern the temporal distribution of k.
Unlike b, k is an important descriptor of relative growth [1], [5]–[7]. In ontogenetic studies of allometry, k is the coefficient of interest because it describes the specific growth rate of 
relative to the specific growth rate of 
[1], [5]–[7]. Thus, the dimensionless ontogenetic allometry coefficient, k, is commonly referred to as the relative growth rate. Since allometric growth is inherently a stochastic process, k must be defined via stochastic analysis; but before this is done, it is necessary to first discuss important mathematical concepts, definitions, and notations used throughout this paper.
The mathematical analysis of k
Let 
and 
each be a deterministic ontogenetic growth function such that 
and 
are deterministic variables that depend on t. Also, let 
be the ratio of 
to 
. Then the first-order derivative of the deterministic ontogenetic growth function 
with respect to the deterministic ontogenetic growth function 
is [1], [5]–[7]
where 
and 
are differentiable real-valued functions of t. Note: 
is a parametric derivative in which 
and 
are differentiable deterministic functions. The temporal distribution of k has been a subject of intense interest (see [6] and [7]). The reason for this is that ontogenetic processes govern the size of traits and the timing and rate of development of traits [7]–[11]. Thus, k can vary with t [5]–[7]; this implies that the relationship between 
and 
may not always be linear [5]–[7]. When 
and 
are linearly related, 
is proportional to 
[1]; k is constant with t, and so the relationship between 
and 
is described by 
. In contrast, when 
and 
are nonlinearly related, 
is not proportional to 
[5]; k varies with t, and so the relationship between 
and 
is not described by 
. Both cases have been observed experimentally (see [12] and [13]). Although deterministic log-linear allometric growth trajectories are always the result of 
being proportional to 
, the proportionality between 
and 
is not always expected to hold under stochastic log-linear allometric growth trajectories because 
and 
are correlated stochastic processes; their probability distributions interact in ways that are not intuitively obvious. The following is a case in point.
Since 
and 
are inherently correlated stochastic processes, 
contains the central and mixed moments of those processes (Methods, equations 6–8). These statistical moments are described by the allometric growth factors (see Methods, equation 9) that affect the temporal distribution of 
. Of course, 
must be transformed into its probabilistic derivative, 
, in order to analyze the allometric growth factors that affect the temporal distribution of k. These allometric growth factors, which only appear in the probabilistic version of 
, are essential because they provide important insight into ontogenetic allometry. Failure to account for the inherent stochasticity in 
leads not only to the miscalculation of k, but also to the omission of all of the informative central and mixed moments of the random ontogenetic growth functions that govern the statistical dynamics of k. Therefore, by treating 
and 
as correlated stochastic processes, this study reveals and analyzes the allometric growth factors that affect the temporal distribution of k.
The probabilistic derivative, 
, in which 
is a ratio of correlated stochastic processes, is newly presented in this study as the inner mean derivative of a random function with respect to a random function. This derivative implies the differentiation of the expected value of a random function with respect to the expected value of a random function, whereas the outer mean derivative of a random function with respect to a random function—for instance, 
—implies the expected value of a ratio of correlated stochastic t-derivatives. In other words, 
, in which 
is a stochastic process, defines k as a deterministic variable, whereas 
, in which 
and 
are stochastic, is the deterministic coefficient 
. Although all of the statistical moments of k can be derived from 
, 
or 
for any 
cannot vary with t because 
is simply a random variable, not a stochastic process. Thus, only 
, by which the deterministic variable k is defined, can vary with t. This distinction between the inner mean derivative 
and the outer mean derivative 
is important and is further addressed in the Discussion.
The concept of an inner mean derivative and an outer mean derivative only applies to the ratio of stochastic t-derivatives. The expected value of a stochastic t-derivative, such as 
, is simply referred to as a mean t-derivative (see equation 4.62 in [14]). Nelson [15] introduced mean derivatives (albeit based on the conditional expectation) to address issues in stochastic mechanics (see [16] and [17] for details).
The probabilistic version of 
is not readily calculable because the numerator and the denominator of 
are correlated stochastic processes; the expected value of a ratio of correlated stochastic processes is generally not equal to the ratio of expected values of the stochastic processes [18]. Therefore, this study equates 
to its Taylor series expansion in order to reveal the central and mixed moments of the stochastic processes on which 
operates (Methods, equations 6–8). Although 
can be expanded as 
(which is not the Taylor series for 
), 
, like its identity 
, is not readily calculable because 
is stochastic. Subsequently, the Taylor series expansion of 
is essential for evaluating the probabilistic version of 
. Also, 
contains the term 
, which is the ratio of 
to 
(Methods, equation 8). Naturally, 
and 
share similar statistical properties; for example, 
equals zero at every t, and 
equals 
for every 
.
Results: The statistical dynamics of k
Using the definitions and notations described above, the inner mean derivative of the random ontogenetic growth function 
with respect to the random ontogenetic growth function 
is (see Methods, equations 6–11, for derivation)
(1)where 
and 
are differentiable real-valued functions of t. Equation (1) is the exact probabilistic version of k. This equation is also the exact general solution for the inner mean derivative of a random function with respect to a random function and can thus be applied to any ratio of correlated stochastic t-derivatives; no simplifying assumptions were made to derive equation (1). Note: 
for each 
is a parametric derivative in which 
and 
are differentiable deterministic functions.
Each of the nth terms in equation (1) is the statistical relative growth rate, 
, which can be expanded as (see Methods, equations 12 and 13, for derivation)
(2)where the summed terms
describe the allometric growth factors that affect the temporal distribution of k. The 0th term in equation (1) is 
(where 
at every t and 
at every t), which becomes k either when 
is deterministic or when 
is zero at every t. Traditionally, 
is calculated as k and is the ratio of 
to 
[19]. Note, however, that evaluating only 
when 
is not zero does not yield an exact k because the other terms—
, 
,…,
—must also be considered. Thus neglecting 
clearly leads to a miscalculated k. Moreover, k (or 
for every 
) can vary with t; nonlinear allometries can occur, even though the stochastic process 
and the stochastic process 
are linearly related.
The statistical dynamics of k can be readily analyzed by the summed terms (
, 
, and 
) in equation (2). Consider the following example: let the stochastic processes, 
and 
, belong to the finite family of 
—the exponential growth-law functions in which only r is a random variable—such that the random ontogenetic growth function 
is 
and the random ontogenetic growth function 
is 
. Then, if 
equals zero, equation (2) is (see Methods, equations 14–16, for derivation)
(3)The allometric growth factors in equation (3) are
Equation (3) is an example of equation (2) in which the derivatives are explicitly defined. The appeal of this example (besides that it can be realistic for a particular organism) is that the allometric growth factors (
and 
) contain the slopes (
and 
) from 
and 
, thus making it easy to interpret the biology of 
and 
. For instance, 
is simply 
; it is the ratio of 
(the expected value of the specific growth rate of 
) to 
(the expected value of the specific growth rate of 
). So, naturally, when the mean growth rate of 
increases relative to the mean growth rate of 
, 
also increases. Note that k differs from 
because 
and 
are nonzero sums. If 
is 1 and 
and 
were both zero sums, then relative growth would be isometric [2]; however, since 
and 
are really nonzero sums, relative growth deviates from isometry. This is a simple and yet realistic example illustrating the fact that k can be miscalculated if 
and 
are not taken into account.
The statistical relative growth rate, 
(where 
), in equation (3) is
The nonzero coefficient, 
, is the probability covariance between the random variable 
and the random variable 
; it is a measure of the joint distribution of 
and 
. The more closely 
and 
are positively associated, the lower the value of 
because 
is less than zero. In contrast, the more closely 
and 
are negatively associated, the higher the value of 
because 
is greater than zero. And so whether 
is being subtracted or added by 
solely depends on the direction of association between 
and 
.
The allometric growth factor (
) contains the term 
, which in equation (3) is
Clearly, 
is a random variable, not a stochastic process. Thus, for instance, 
is a nonzero positive coefficient that represents the ratio of 
(the probability variance of 
) to 
(the squared expected value of 
). Consequently, 
describes the ontogenetic variance of 
, and 
describes the ontogenetic asymmetry of 
. Both genetic and environmental factors can affect 
and 
, and these two ontogenetic statistical moments (or biological processes) influence k in a manner that is not intuitively obvious unless equation (1) is used.
It is important to note that the allometric growth factor, 
, is zero in equation (3) only because 
is a random variable, not a stochastic process; 
does not vary with t because 
is constrained to zero, and thus 
equals zero at every t. Since 
is constrained to zero, k does not vary with t.
Now suppose only 
and 
are random variables in the random ontogenetic growth functions, 
and 
. Then, if 
equals zero, equation (2) is (see Methods, equations 17–19, for derivation)
(4)where 
is a deterministic process and 
is a mean-centered random variable. In this case, 
is a stochastic process because 
for each 
does not vary with t, and yet 
increases with t since there is growth. The allometric growth factors in equation (4) are
It is apparent that, unlike equation (3), equation (4) contains the deterministic variable t. Thus, k varies with t, and its values can either be greater than 1 (that is, positively allometric at every t) or less than 1 (that is, negatively allometric at every t) or an arrangement of both (that is, reversal in ontogenetic polarity) [10]. Note that 
in equation (4) is constant with t; this implies that 
and 
are linearly related, and so the relationship between 
and 
is described by 
. All other statistical relative growth rates (
for every 
), however, are derived from relationships that are not described by 
and therefore vary with t. For example, 
and 
are nonlinearly related, and so 
, which is derived from the relationship between 
and 
, varies with t. Consequently, nonlinear allometries occur in this case, even though the stochastic process 
and the stochastic process 
are linearly related.
Intricate temporal distributions of k can arise from the case described by equation (4). For example, suppose 
at every t is negligible compared to 
at every t. Then equation (1) is
where 
and 
are probabilistic coefficients. Now there could be a condition in which the temporal distribution of k is not monotonic and is either positively allometric or negatively allometric: k has a stationary point at 
(set 
and solve for t), where the stationary value of k is 
(substitute 
for t in k); thus, the temporal distribution of k is not monotonic. This is an interesting case because k, which could be either greater than 1 or less than 1 at every t, increases with t, reaches 
(the maximum rate of relative growth), and then decreases with t. This is classic case of accelerated and decelerated rates of relative growth within a given t period. Note that 
depends on the probabilistic coefficients, 
, 
, and 
. When 
is deterministic, 
is undefined. Since, however, 
is inherently stochastic, the terms in 
and in 
affect 
and 
. For instance, if 
increases while 
, 
, and all other terms in 
remain constant, then 
increases, assuming 
and 
are positive. Moreover, increasing 
decreases the t at which 
is reached; this is because 
is inversely proportional to 
, which is directly related to 
. If stochasticity disappears, then 
and 
also vanish and 
becomes undefined. So 
, 
, and 
affect not only 
, but also the t at which 
is reached. This is a clear case of how 
and 
—coefficients that only appear in the probabilistic version of 
—affect the timing and rate of development of traits. Thus, ignoring the effects of stochasticity on both 
and 
omits all of the informative ontogenetic statistical moments (e.g., 
) that govern the temporal distribution of k. Furthermore, even though the relationship between the realizations of stochastic 
and the realizations of stochastic 
is described by 
, k differs from 
and can vary with t. This important fact should always be considered when analyzing allometric growth.
It is interesting to note that as t approaches infinity, equation (4) or any of its approximations reaches an asymptotic value of 
. The t at which this asymptotic value is first reached is an indication of the cessation of the variability of k with respect to t. In other words, nonlinear allometries disappear as t approaches infinity. So as the allometric growth process evolves over t, two distinct phases are observed: the first phase is a non-uniform temporal distribution of k, and the second phase is a uniform temporal distribution of k. This two-phase allometric growth process may be more realistic than a growth process that exclusively corresponds to either the first or second phase. It should be made clear, though, that only the second phase is indicative to a log-linear allometric growth trajectory, since 
(not k) is constant with t. And so the probabilistic coefficients, 
and 
, essentially have an insignificant impact on only the second phase of the allometric growth process. Clearly, the first phase of the allometric growth process can entail an intricate temporal distribution of k, such as the one provided in the previous paragraph.
Equations (3) and (4) are realistic examples of the types of temporal distributions of k that may arise from the random exponential growth-law function, 
, to which the stochastic processes, 
and 
, belong. The important distinction between equations (3) and (4) is the type of variable 
assumes: 
is a random variable (not a stochastic process) in equation (3); 
is a stochastic process in equation (4). As a result, k defined by equation (3) does not vary with t, whereas k defined by equation (4) varies with t. In either case, it is q or r that is a random variable. Nonetheless, it is entirely possible to have a case in which q and r are both random variables.
With regard to the convergence of equation (1), 
has an important role: equation (1) is guaranteed to converge at every t if the realizations of stochastic 
are between −1 and 1 at every t; this is because the realizations of 
approach zero at every t as n approaches infinity.
Discussion
Although statistical models for relative growth have been developed (see [7] and [20]), their models, which show variability in 
with respect to t, are not probabilistic because they do not incorporate actual stochasticity into 
; they do not treat 
and 
as correlated random functions. Also, although a probabilistic model for static (not ontogenetic) allometry, in which x is treated as an independent random variable (not as a stochastic process), has been proposed (see [21]), their model cannot address the statistical moments that govern the temporal distribution of k because their model is used to analyze the effects of stochasticity only on 
. Consequently, equation (1) is entirely new and has no analog to any statistical model for relative growth previously developed.
Equation (1) is the exact general solution for the inner mean derivative of the random ontogenetic growth function 
with respect to the random ontogenetic growth function 
. This equation, which is the exact probabilistic version of k, is general because it does not entail any simplifying assumptions. Thus, the generality of equation (1) makes it possible to analyze all of the informative ontogenetic statistical moments (or biological processes) that govern the temporal distribution of k:
This expression makes it apparent that k is composed of an infinite series of ratios of first-order t-derivatives. The statistical complexity of k arises from the derivative in the numerator, which is the t-derivative of the nth mixed moment of 
and 
. Each of these nth statistical moments is governed by the interactions between 
and 
. So most of the informative ontogenetic statistical moments are captured by the mean t-derivative, 
; this is evident by expanding 
(see Methods, equations 12 and 13, for derivation):
(5)
The summed terms in equation (5) compose the allometric growth factors (
, 
, and 
) in equation (2). These allometric growth factors are important to interpret because they describe the central and mixed moments of the random ontogenetic growth functions that govern the statistical dynamics of k. Clearly, equation (5) is calculable, since each of the nth terms of 
is a differentiable deterministic function of t.
To biologically interpret equation (5), one must specify the finite family of functions to which the stochastic processes, 
and 
, belong (see, for example, equations 3 and 4).
Equations (3) and (4) are examples of how to model and analyze the statistical dynamics of k. These examples are derived from the random exponential growth-law function that is theoretically assumed for a particular organism. Thus relaxing this assumption leads to the practical (experimental) side of modeling the statistical dynamics of k. Traditionally, 
and 
are experimentally measured, plotted with respect to each other, and then related by a differentiable function from which 
is derived [19]. This study, however, shows that 
is not the only statistical relative growth rate that needs to be considered when evaluating k (see equations 1 and 2). The other statistical relative growth rates (
for every 
) should also be quantified in a similar manner. For example, 
and 
can be experimentally measured, plotted with respect to each other, and then related by a differentiable function from which 
can be derived. Thus, the probabilistic version of 
is a very practical metric: it only requires measuring the mixed and central moments of 
and 
.
The ontogenetic growth functions, 
and 
, must be linearly related in order to satisfy the log-linear allometric function, 
. Thus, 
and 
can be generalized as 
and 
, where deterministic or stochastic 
is any differentiable function of t. In equations (3) and (4), 
is simply t; but, to describe more intricate ontogenetic growth distributions, 
could also be 
for any 
, where 
for each 
is a deterministic or stochastic parameter. Note that 
and 
equals 
and 
, respectively; this is true for any distribution of 
. Subsequently, 
equals 
, which is the expected value of the ratio of 
to 
.
For most organisms, 
is constant with t; this implies that 
and 
are typically zero at every t. In equations (3) and (4), where 
is t, 
and 
are naturally zero because t is naturally deterministic; thus, 
is naturally constant with t in these equations. There are some organisms (predominately plants) that show 
varying with t [22]. Indeed, this case, in which 
varies with t, is interesting to study, but complicates the biological analysis of 
because the biological interpretation of 
or 
cannot explicitly be defined. Therefore, when analytically modeling 
, there is good reason to assume that 
and 
are zero at every t. Keep in mind, though, that while stochastic 
and stochastic 
are linearly related, 
can vary with t.
It is important to note that if 
is not a stochastic process, then k (which differs from 
) does not vary with t (see equation 3). If, however, 
is a stochastic process, then k not only differs from 
, but also varies with t (see equation 4); this implies that the statistical relative growth rates (
for every 
) are derived from relationships that are not described by 
, even though the stochastic process 
and the stochastic process 
are linearly related.
Another important point to note is that 
is mathematically different from the expected value of a ratio of correlated stochastic t-derivatives. If 
and 
are correlated stochastic t-derivatives, then the outer mean derivative, 
, is generally not identical with equation (1). Stated more explicitly,
and
are generally not identical with equations (1) and (2), respectively. Note: 
and 
are derived in exactly the same manner as 
(see Methods, equation 8) and 
(see Methods, equation 9). Now compare the following limits: the outer mean derivative is
whereas the inner mean derivative is
Thus, in 
, the limit operates on the ratio of stochastic 
to stochastic 
; but in 
, the limit operates on the ratio of deterministic 
to deterministic 
. So 
is identical with 
when both 
and 
are deterministic or when only 
is stochastic. When, however, only 
is stochastic or when both 
and 
are stochastic, 
is generally not identical with 
(see equation 4); the only exception is the special case when 
is not a stochastic process, but a random variable (see equation 3). As a result, the outer mean derivative 
is a special case of the inner mean derivative 
. Also, 
is equal to 
.
In conclusion, equation (1) is completely versatile and has much to offer with regard to analyzing the allometric growth factors (
, 
, and 
) that affect the temporal distribution of k. When the derivatives in equation (2) are defined explicitly via specifying the random ontogenetic growth functions (
and 
), the allometric growth factors become biologically interpretable; they also become tractable in simulations, which are useful for modeling the statistical rates of relative growth for various distributions of 
(see Methods, Simulating the probabilistic version of k). Thus, each of the statistical relative growth rates (
, 
,…, 
), which are infinitely summed to form equation (1), can be analyzed in detail to reveal new insight into the statistical dynamics of relative growth.
Lastly, this study ignored the statistical dynamics of b because only k is an important descriptor of relative growth. But to obtain a complete characterization of the statistical dynamics of allometric growth, b or 
must also be considered. Since the stochastic analysis of k has been fully developed in this study (see Methods, equations 6–11), the exact probabilistic version of β can easily be formulated:
where 
is the ratio of 
to 
and 
is the ratio of 
to 
. Each of the nth terms of 
is the allometric growth descriptor, 
:
The summed terms in 
describe the allometric growth factors that affect the temporal distribution of β. The equation (
) contains all of the ontogenetic statistical moments that govern the temporal distribution of β. And just like k, one could analyze the statistical dynamics of β simply by examining the summed terms in 
. Note that, like 
in equation (1), if 
is a stochastic process, then β varies with t.
Methods: The stochastic analysis of k
Let 
and 
each be a random ontogenetic growth function such that 
and 
are correlated stochastic processes. Then, if 
is the ratio of 
to 
, the expected value of 
is
(6)Equation (6) contains the central and mixed moments of 
and 
. These statistical moments can be revealed by expanding equation (6) using the Taylor series generated by the function, 
, defined by the denominator 
when α equals zero at every 
:
(7)where 
is the ratio of 
to 
. Substituting equation (7) into equation (6) yields
(8)where each of the nth terms in equation (8) is 
:
(9)The summed terms in equation (9) are the allometric growth factors that affect the temporal distribution of 
. Rice and Papadopoulos [23] use a similar mathematical approach (that is, the Taylor series expansion of the expected value of the change in mean phenotype) to reveal important biological factors governing evolution.
Equation (8), which is the Taylor (or Maclaurin) series expansion of 
, can also be expressed as 
. This particular expression, however, has no explicit common denominator, as its denominator has an unfixed exponent; thus, 
cannot operate linearly on this expression, and consequently fails to define k from this expression. In contrast, equation (1), in which 
operates specifically on equation (8), uniquely defines the probabilistic version of k. Equation (8) is thus essential for evaluating 
: the t-derivative of the numerator in equation (8) is
(10)and the t-derivative of the denominator in equation (8) is
(11)Therefore, equation (1) (that is, the inner mean derivative of the random ontogenetic growth function 
with respect to the random ontogenetic growth function 
) is the ratio of equation (10) to equation (11):
Now the identity 
can be used to expand 
:
(12)the product rule is used to expand 
:
(13)Substituting equation (13) into equation (12) and dividing by 
yields the expanded form of the statistical relative growth rate, 
:
which is identical with equation (2). The summed terms in equation (2) are the allometric growth factors (
, 
, and 
) that affect the temporal distribution of k:
and
When 
is a stochastic process, the product or quotient rule can be used in 
and in 
to calculate their derivatives. Note that 
and 
represent deterministic t-derivatives of the product of two deterministic functions.
Now suppose for a particular organism the random ontogenetic growth functions, 
and 
, are defined by 
and 
in which only 
and 
are random variables. Then the allometric growth factors, which are the summed terms in equation (2), are as follows:
(14)
(15)and
(16)where 
is a random variable, not a stochastic process. Summing equations (14), (15), and (16) then yields equation (3):
If, however, 
and 
are defined by 
and 
in which only 
and 
are random variables, then the allometric growth factors are
(17)
(18)and
(19)where 
is a stochastic process and 
is a mean-centered random variable. Summing equations (17), (18), and (19) then yields equation (4):
Methods: Simulating the probabilistic version of k
Simulating 
using 
and 
as correlated random functions can easily be done: first specify the terms in 
and in 
that are stochastic and then provide their (joint) probability distributions. Because the stochastic process 
and the stochastic process 
are linearly related and because 
and 
are assumed to be zero at every t, 
is constant with t. Thus, the parametric derivative, 
, is readily calculable, since 
and 
are known from the distribution of 
and the distribution of 
, respectively. In contrast, 
for each 
is not readily calculable, but can easily be assessed in simulations by first evaluating 
for each 
and then relating 
to 
by a differentiable function from which the derivative (i.e., 
) can be calculated. So, for example, 
is the parametric derivative, 
; to evaluate 
properly in simulations, the following identity of 
should be used: 
; this is because 
is evaluated together with (not separate from) 
and 
in simulations. Therefore, the binomial expansion of 
is useful for numerically evaluating 
:
