Generalized Gaussian Laws

B. F. Krutoi

(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)

Let us denote by

\[ \operatorname{Pr}(\xi=x)=\varphi(x) \]

the probability that the measured value of the independent random variable \(\xi\) will be equal to \(x\). This probability is in good agreement with experiment if it is calculated in accordance with the exact Gaussian law [1]

\[ \operatorname{Pr}(\xi=x)=\varphi(x)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-c)^2}{2\sigma^2}}. \tag{1} \]

The Gaussian law (1) is represented by the curve shown in Fig. 1.

Fig. 1. Curves of Gaussian laws: ordinary (I) and generalized (II)

The quantities \(c\) and \(\sigma\) entering into (1) are parameters established by the limiting equalities

\[ c=C\xi=\lim_{n\to\infty}\frac{\sum_{i=1}^{n}\tilde{x}_i}{n}, \qquad \sigma^{2}=C[\xi-C\xi]^2=\lim_{n\to\infty}\frac{\sum_{i=1}^{n}(\tilde{x}_i-c)^2}{n}, \tag{2} \]

where \(C\) is the symbol for the mean value of a variable quantity, and \(\tilde{x}_i\) is one of

possible values of the random variable \(\xi\). The parameter \(c\) is called the mean value of the random variable \(\xi\), and the parameter \(\sigma\) the mean dispersion of the random variable \(\xi\).

In actual observations, instead of the possible values \(\tilde{x}_i\) of the random variable \(\xi\), we obtain its measured values \(x_i\), and the number of observations \(n\) is always limited. In this case the parameters \(c\) and \(\sigma\), which enter into (1) and are defined by the limiting equalities (2), are replaced by their observed values \(c_{\mathrm{obs}}=\bar{x}\) and \(\sigma_{\mathrm{obs}}=m\), which are found as follows:

\[ c_{\mathrm{obs}}=\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i,\qquad \sigma_{\mathrm{obs}}^{2}=m^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^{2}. \tag{2a} \]

Instead of (1) we then obtain, approximately,

\[ \operatorname{Prob}(\xi=x)=\varphi(x)\simeq \frac{1}{m\sqrt{2\pi}}\,e^{-\frac{(x-\bar{x})^{2}}{2m^{2}}}. \tag{1a} \]

We shall further denote by

\[ \operatorname{Prob}(x_1\leq \xi \leq x_2)=\Phi(x_1,x_2) \]

the probability that the measured value of the independent random variable \(\xi\) will be contained within the limits \((x_1\leq \xi \leq x_2)\), where \(x_1,x_2\) are numbers specified in advance. Then calculations show that if \(\operatorname{Prob}(\xi=x)\), with \(x\) variable, follows the pointwise Gaussian law (1), then the probability of finding \(\xi\) within the limits \((x_1\leq \xi \leq x_2)\) is determined by the integral Gaussian law

\[ \operatorname{Prob}(x_1\leq \xi \leq x_2)=\Phi(x_1,x_2)= \frac{1}{\sigma\sqrt{2\pi}}\int_{x_1}^{x_2} e^{-\frac{(z-c)^{2}}{2\sigma^{2}}}\,dz, \tag{3} \]

where the variable \(x\) has been denoted by \(z\). Substituting into (3), instead of the parameters \(c,\sigma\), their observed values \(\bar{x}=z\) and \(m\), computed according to (2a), we shall have, instead of (3), the following approximate equality:

\[ \operatorname{Prob}(x_1\leq \xi \leq x_2)=\Phi(x_1,x_2)\simeq \frac{1}{m\sqrt{2\pi}}\int_{x_1}^{x_2} e^{-\frac{(z-\bar{z})^{2}}{2m^{2}}}\,dz. \tag{3a} \]

We shall consider the question of computing integrals of the form (3) or (3a) somewhat later.

Let us clarify the origin of the factor \((\sigma\sqrt{2\pi})^{-1}\) entering into the basic equalities (1) and (3). For this purpose let us recall certain features of Gauss’s derivation of his integral relation (3), [2].

In deriving equality (3), Gauss starts from \(\operatorname{Prob}(\xi=x)\), given—

in a more general form than (1), namely

\[ \operatorname{Prob}(\xi=x)=\varphi(x)=Ae^{-\frac{(x-c)^2}{2\sigma^2}}, \tag{1б} \]

where the factor \(A\) is an as yet unknown quantity, determined by some additional condition. Therefore, instead of (3), we initially have for \(\operatorname{Prob}(x_1 \le \xi \le x_2)\) the following general expression:

\[ \operatorname{Prob}(x_1 \le \xi \le x_2)=\Phi(x_1,x_2)= A\int_{x_1}^{x_2} e^{-\frac{(z-c)^2}{2\sigma^2}}\,dz. \tag{3б} \]

The factor \(A\), entering into (3б), Gauss finds from the consideration that for \(x_1=-\infty,\ x_2=+\infty\)

\[ \operatorname{Prob}(-\infty \le \xi \le +\infty)=\Phi(-\infty,+\infty)=1. \]

Hence, for determining the unknown \(A\), we obtain the very simple equation

\[ A\int_{-\infty}^{+\infty} e^{-\frac{(z-c)^2}{2\sigma^2}}\,dz=1. \tag{a} \]

Introducing here the substitution

\[ \frac{z-c}{\sigma}=t \]

and taking into account the evenness of the integrand, we shall have, instead of \((a)\),

\[ 2\sigma A\int_{0}^{\infty} e^{-\frac{t^2}{2}}\,dt=1. \tag{б} \]

But, as is known,

\[ \int_{0}^{\infty} e^{-\frac{t^2}{2}}\,dt=\frac{\sqrt{2\pi}}{2}. \]

Therefore, for the factor \(A\), Gauss obtains from (б) the following final value:

\[ A=\frac{1}{\sigma\sqrt{2\pi}}=A_1, \tag{4} \]

which agrees with (1) and (3). At the same time, from (1) it follows that, for

\[ x_1=-\infty,\quad x_2=+\infty \]

\[ \varphi(-\infty)=\varphi(+\infty)=e^{-x^2}=0, \tag{1в} \]

that is, both branches of the Gaussian curve (1) touch the \(X\)-axis at infinity (Fig. 1).

Let us discuss the indicated feature of the Gaussian curve (1). The circumstance that the point law of Gauss (1), corresponding to this curve, gives for \(\operatorname{Ver}(\xi=x)\) the value \(0\) only when \(x=+\infty\), contradicts our conception of the actual limits of possible values \(\tilde{x}_i\) of the independent random variable \(\xi\) under given conditions of its measurement. Extensive production experience convinces us that, in the absence of gross errors, our measurements give for the independent random variable \(\xi\) a series of observed values \(x_i\) which do not go beyond rather narrow limits \(\lambda_1 < x_i < \lambda_2\), determined by the precision of the instruments and the influence of the external environment. For example, measuring a line length \(d=1000.0\ \text{m}\) under average conditions with a 20-meter tape with pins, we cannot admit that, in the absence of gross mistakes, the measured values \(d_i\) of the length \(d\) can reach the limiting value \(\lambda_1=995.0\ \text{m}\) or \(\lambda_2=1005.0\ \text{m}\). In the same way, when measuring a horizontal angle \(\gamma=75^\circ18'35''\) with a theodolite with graduation value of the reading device \(\mu=2''\) on the horizontal circle, it is impossible to suppose that, in the absence of gross miscalculations and under average visibility conditions, the measured values \(\gamma_i\) can reach the limiting value \(\lambda_1=75^\circ18'25''\) or \(\lambda_2=75^\circ18'45''\).

From the considerations presented it follows beyond doubt that, under actual conditions of measurement of the independent random variable \(\xi\), the equalities

\[ \operatorname{Ver}(\xi=\lambda_1)=0,\qquad \operatorname{Ver}(\xi=\lambda_2)=0 \]

will be quite reliable for values \(\lambda_1,\lambda_2\) that differ strongly from \(-\infty,+\infty\). This property of actual measurements suggests replacing the well-known Gaussian laws, the point law (1) and the integral law (3), by laws close to them, but more correct. The chief feature of these new laws, which we shall call generalized Gaussian laws, will consist in the fact that in them there are established, much more precisely than by Gauss himself, those limits \(\lambda_1,\lambda_2\) which the possible values \(\tilde{x}_i\) of the independent random variable \(\xi\) certainly will not reach under actual measurement conditions. By this means we shall obtain a more accurate estimate of the degree of reliability of the measured values \(x_i\) of the random variable \(\xi\). Below, the derivation of the corresponding generalized Gaussian laws is given.

Let \(\lambda_1,\lambda_2\) be the above-mentioned limits of the possible values \(\tilde{x}_i\) of the independent random variable \(\xi\) subject to our measurement. Hence, under the given measurement conditions, the equalities

\[ \operatorname{Ver}(\xi=\lambda_1)=\psi(\lambda_1)=0,\qquad \operatorname{Ver}(\xi=\lambda_2)=\psi(\lambda_2)=0 \tag{5} \]

are entirely exact; only the form of the function \(\psi\) remains for the present undetermined. Somewhat refining the form of the function \(\psi\), we shall assume, in accordance with (1) and (3), that

\[ \operatorname{Ver}(\xi=x)=\psi(x) \tag{6} \]

\[ \operatorname{Ver}(x_1\leq \xi \leq x_2)=\Psi(x_1,x_2)=\int_{x_1}^{x_2}\psi(z)\,dz, \tag{7} \]

where \(\psi(x)\) is a continuous and differentiable function of \(x\). In addition, the function \(\psi(x)\) must have the property that, for certain experimentally established values \(x_1=\lambda_1\) and \(x_2=\lambda_2\), the equality

\[ \operatorname{Prob}(\lambda_1 \leqslant \xi \leqslant \lambda_2) = \Psi(\lambda_1,\lambda_2) = \int_{\lambda_1}^{\lambda_2} \psi(z)\,dz = 1 \tag{8} \]

is perfectly exact. The significance of this condition is obvious.

Wishing subsequently to obtain probability distribution laws of the independent random variable \(\xi\) that are close to the Gaussian ones (1), (3) and satisfy condition (8), we shall choose the limits \(\lambda_1\) and \(\lambda_2\) in such a way that

\[ -(\lambda_1-c)=(\lambda_2-c), \tag{9} \]

where \(c=C\xi\) according to (2). Taking (9) into account, let us now define the function \(\psi(x)\) by the relation

\[ \psi(x)=\operatorname{Prob}(\xi=x)=\varphi(x)-\varphi(\lambda_1) =\varphi(x)-\varphi(\lambda_2), \tag{10} \]

in which the function \(\varphi(x)\) is established by equality (16), expressing the point Gaussian law in a somewhat generalized form. Therefore, instead of (10) we shall have the more expanded form

\[ \operatorname{Prob}(\xi=x)=\psi(x) = A e^{-\frac{(x-c)^2}{2\sigma^2}} - A e^{-\frac{(\lambda_1-c)^2}{2\sigma^2}} = \]

\[ = A e^{-\frac{(x-c)^2}{2\sigma^2}} - A e^{-\frac{(\lambda_2-c)^2}{2\sigma^2}}, \tag{10a} \]

where \(A\) is, for the time being, an undetermined factor. It is easy to verify that the function \(\psi(x)\) specified by this expression also satisfies the boundary conditions (5).

Relation (10a), for \(A=(\sigma\sqrt{2\pi})^{-1}=A_r\), evidently gives the same Gaussian curve as equation (1), but only refers this curve to a new axis \(X'\), passing through the points with coordinates \([\lambda_1,\varphi(\lambda_1)]\), \([\lambda_2,\varphi(\lambda_2)]\) in the old reference frame \(XY\) (Fig. 1). Since the limits \(\lambda_1,\lambda_2\) were chosen by us under condition (9), \(\varphi(\lambda_1)=\varphi(\lambda_2)\), and therefore the new axis \(X'\) is parallel to the old axis \(X\), but is situated somewhat above it.

Taking the foregoing into account, let us refer the Gaussian curve (1) to the new reference frame \(X'Y\), for which, according to (5),

\[ \psi(\lambda_1)=\psi(\lambda_2)=0, \]

and consider the part of this curve contained between the points \((\lambda_1,0)\), \((\lambda_2,0)\) of its intersection with the axis \(X'\). Since the Gaussian curve (1) is tangent to the original axis \(X\) at the points \(+\infty\), \(-\infty\) and forms with this axis an area equal to \(1\)—condition (a)—the above-indicated part of the Gaussian curve (1) will form with the corresponding segment \([\lambda_1,\lambda_2]\) of the new axis \(X'\) an area less than \(1\) (Fig. 1). Meanwhile, proceeding from the actual conditions of measurement, we have established that, independently of the distribution law \(\psi(x)\) of the independent random variable \(\xi\),

\[ \operatorname{Prob}(\lambda_1 \leqslant \xi \leqslant \lambda_2) = \Psi(\lambda_1,\lambda_2) = \int_{\lambda_1}^{\lambda_2} \psi(x)\,dx = 1. \tag{8} \]

Assuming, therefore, that condition (8) is also satisfied by that particular form of the function \(\psi(x)\) which we specified by relation (10a), we find from condition (8) the undetermined multiplier \(A\) entering into (10a). We shall have

\[ \operatorname{Ver}(\lambda_1 \leq \xi \leq \lambda_2) = \Psi(\lambda_1,\lambda_2) = A\left( \int_{\lambda_1}^{\lambda_2} e^{-\frac{(z-c)^2}{2\sigma^2}}\,dz - e^{-\frac{(\lambda_1-c)^2}{2\sigma^2}} \int_{\lambda_1}^{\lambda_2} dz \right) =1, \]

whence, for calculating \(A\), we obtain the following final expression

\[ A= \left[ \int_{\lambda_1}^{\lambda_2} e^{-\frac{(z-c)^2}{2\sigma^2}}\,dz - (\lambda_2-\lambda_1) e^{-\frac{(\lambda_1-c)^2}{2\sigma^2}} \right]^{-1}. \tag{11} \]

Introducing here the substitutions

\[ \frac{z-c}{\sigma}=t,\qquad \frac{\lambda_1-c}{\sigma}=-k \tag{12} \]

and taking (9) into account, we reduce (11) to a form more convenient for computations,

\[ A= \left[ 2\sigma\left( \int_{0}^{k} e^{-\frac{t^2}{2}}\,dt - k e^{-\frac{k^2}{2}} \right) \right]^{-1}. \tag{13} \]

Expression (13), with the notation (12) taken into account, shows that the multiplier \(A\) entering into (10a) depends on three parameters: \(c,\sigma,\lambda_1\). For \(\lambda_1=-\infty\) this multiplier becomes \(A_1=(\sigma\sqrt{2\pi})^{-1}\). The calculation of the integral contained in (13) will be considered below.

Having specified the multiplier \(A\) entering into expression (10a) for the function \(\psi(x)\), we insert this expression for the function \(\psi(x)\) into the original equalities (6), (7). Then we obtain two transformed equalities:

\[ 1)\quad \operatorname{Ver}(\xi=x)=\psi(x) = A\left( e^{-\frac{(x-c)^2}{2\sigma^2}} - e^{-\frac{(\lambda_1-c)^2}{2\sigma^2}} \right). \tag{14} \]

\[ 2)\quad \operatorname{Ver}(x_1\leq \xi \leq x_2) = \Psi(x_1,x_2) = \]

\[ = A\left( \int_{x_1}^{x_2} e^{-\frac{(z-c)^2}{2\sigma^2}}\,dz - (x_2-x_1) e^{-\frac{(\lambda_1-c)^2}{2\sigma^2}} \right). \tag{15} \]

Since the indicated transformed equalities were found by an appropriate modification of Gauss’s laws (1), (3), these equalities (14), (15) are precisely those generalized Gauss laws, point and integral, which we wished to find. The factor \(A\) entering here is determined according to (13). It is easy to see that, for \(\lambda_1=-\infty\), \(\lambda_2=+\infty\), these generalized laws (14), (15) turn into the ordinary ones (1), (3).

The generalized Gauss laws (14), (15) can be given a more concise and, at the same time, more convenient form for computations if they are written as follows:

\[ \text{1) } \operatorname{P}(\xi=x)=\psi(x)=A\left(e^{-\frac{t^2}{2}}-e^{-\frac{k^2}{2}}\right). \tag{14a} \]

\[ \text{2) } \operatorname{P}(x_1\leq \xi \leq x_2)=\Psi(x_1,x_2)= \]

\[ =A\left[\sigma\left(\int_0^{h_2} e^{-\frac{t^2}{2}}\,dt-\int_0^{h_1} e^{-\frac{t^2}{2}}\,dt\right) -(x_2-x_1)e^{-\frac{k^2}{2}}\right]. \tag{15a} \]

Here

\[ h_1=\frac{x_1-c}{\sigma}, \qquad h_2=\frac{x_2-c}{\sigma}, \tag{16} \]

and \(t\), \(k\), and \(A\) are determined by relations (12), (13); moreover, in order to shorten the notation we have put

\[ t=\frac{z-c}{\sigma}=\frac{x-c}{\sigma}. \]

As for integrals of the form

\[ \int_0^h e^{-\frac{t^2}{2}}\,dt, \tag{17} \]

there are detailed tables for them, compiled according to the argument \(h\). In addition, integral (17) can be computed directly by means of the expansions [3]

\[ \text{1) } \int_0^h e^{-\frac{t^2}{2}}\,dt =\sqrt{2}\int_0^r e^{-s^2}\,ds= \]

\[ =\sqrt{2}\left(r-\frac{r^3}{3}+\frac{r^5}{5\cdot 2!} -\frac{r^7}{7\cdot 3!}+\frac{r^9}{9\cdot 4!}-\cdots\right). \tag{18} \]

2)

\[ \int_{0}^{h} e^{-\frac{t^{2}}{2}}\,dt = \sqrt{2}\int_{0}^{r} e^{-s^{2}}\,ds = \]

\[ =\sqrt{2\pi} -\frac{e^{-r^{2}}}{r\sqrt{2}} \left( 1-\frac{1}{2r^{2}}+\frac{1\cdot 3}{(2r^{2})^{2}} -\frac{1\cdot 3\cdot 5}{(2r^{2})^{3}}+\cdots \right), \tag{19} \]

where

\[ r=\frac{h}{\sqrt{2}}. \tag{20} \]

Series (18) gives rapid convergence for \(r<1\) and provides computation of the integral (17) with any degree of accuracy, but for \(r>1.5\) the convergence of this series begins to slow down noticeably. Therefore for \(r>2\) one already uses series (19), whose terms decrease until their number exceeds \((r^{2}+1)\). For these reasons one should not take many terms of series (19).

Let us note that if the limits \(x_1, x_2\) of the integral in equality (15) are chosen in such a way that

\[ -\frac{x_1-c}{\sigma}=\frac{x_2-c}{\sigma}=h, \tag{16a} \]

then instead of (15a) we shall have the simpler expression

\[ \operatorname{Prob}(x_1\leq \xi \leq x_2)=\Psi(x_1,x_2)= \]

\[ =2A\sigma \left( \int_{0}^{h} e^{-\frac{t^{2}}{2}}\,dt - h e^{-\frac{h^{2}}{2}} \right). \tag{15b} \]

Having finished the derivation of the generalized Gaussian laws (14), (15), let us consider the question of the extent to which the probabilities calculated from these generalized laws for certain \(x\) and \(x_1,x_2\) differ from the corresponding probabilities found according to the ordinary Gaussian laws (1), (3). For this purpose, using equalities (14a) and (15a), we shall calculate the probabilities \(\psi(x)\) and \(\Psi(x_1,x_2)\) for \(\sigma=2\), for \(k=3,4,5\), and for values of the argument \(h_1=h_2=h=0;\ 0.5;\ 1.0;\ldots;\ 4.5;\ 5.0\). For comparison we shall also compute, according to (1) and (3), the values of the probabilities \(\varphi(x)\) and \(\Phi(x_1,x_2)\) for \(\sigma=2\) and for the same \(h\). Finally, we shall find the corresponding differences \(\Delta\psi(x)\) and \(\Delta\Psi(x_1,x_2)\), where

\[ \Delta\psi(x)=\psi(x)-\varphi(x),\qquad \Delta\Psi(x_1,x_2)=\Psi(x_1,x_2)-\Phi(x_1,x_2). \tag{21} \]

The result of the computations is presented in Tables 1 and 2.

Examination of these two tables shows that for \(k=5\) the functions \(\psi(x)\) and \(\Psi(x_1,x_2)\) coincide, to the fifth decimal place, with the corresponding Gaussian functions \(\varphi(x)\) and \(\Phi(x_1,x_2)\). Hence it follows that, for a reliable estimate of the probabilities of the measured values \(x_i\) of the independent random variable \(\xi\), it is sufficient to take \(k=4\) and even, perhaps, \(k=3\). In actual measurements, in equalities (14), (15) and their derivatives we take, instead of the parameters \(c,\sigma\), their observed values \(\bar{x}\) and \(m\), determined according to (2a).

Table 1

Values of the functions $\psi_k(x)$ and $\psi_\infty(x)=\varphi(x)$
$(\sigma=2)$

Table 2

Values of the functions \(\Psi_k(x_1,x_2)\) and \(\Psi_\infty(x_1,x_2)=\Phi(x_1,x_2)\)
\((q=2)\)

In conclusion, we note that replacing the ordinary Gaussian laws (1), (3) by the generalized laws (14), (15) makes it possible to obtain more accurate estimates of the probabilities

\[ \operatorname{Prob}\left(\frac{\bar{x}}{c}<y\right), \qquad \operatorname{Prob}\left(\frac{m}{\sigma}<z\right). \]

compared with those that we make using Student’s distribution and the \(\chi^2\) distribution [4].

REFERENCES

1. A. A. Markov. Calculus of Probabilities. GIZ, 1924.
2. C. F. Gauss. Werke. Theoria motus. Bd. VII. Gotha, 1871.
3. J. Scarborough. Numerical Methods of Mathematical Analysis. ITTI, 1934.
4. B. V. Gnedenko. A Course in Probability Theory. GITTL, 1950.