A simple approximation to the Gaussian distribution

Structural Safety, 9 (1991) 315-318

315

Elsevier

SHORT COMMUNICATION

A SIMPLE APPROXIMATION TO THE GAUSSIAN DISTRIBUTION Mario Ordaz Instituto de Ingenier~a, UNAM, Ciudad Universitaria, Coyoac~n 04510, DF, Mexico

We present a very simple approximation to the standard Gaussian distribution. While it is not as accurate as others that can be found in the literature [1-3], its form is especially suitable to solve some integrals, involving Gaussian cumulative density functions, that do not have analytical solutions. Let

*(x)=

x

~1

f l o~e

-

u2/2

du

An approximation to ~ ( x ) for nonpositive x is given by

• (x)----0.6931exp{- (914 8)2}

(1)

and since ~ ( - x ) = 1 - ~ ( x ) , eqn. (1) can also be used for x > 0. The absolute errors associated with this estimation are presented in Fig. 1 for the range - 4 ~< x ~< 4, whereas Fig. 2 depicts the relative errors in the same range. The former are acceptable for the whole interval of x and the

A B S O L U T E E R R O R I N ,~(x)

RELATIVE ERROR IN #(x) 10

.004 .002

5

cX 000

0

-.002

-5

5

-.004 I

I

I

~

I

1

-3

-2

-I

0

I

2

-10 t

-3

.T

Fig. 1. Absolute error in computing ~ ( x ) approximation given in eqn. (1).

I

-2

I

-1

I

I

t

I

0

1

2

3

4

X

with the

Fig. 2. Relative error in computing ~ ( x ) approximation given in eqn. (1).

0167-4730/91/$03.50 © 1991 - Elsevier Science Publishers B.V.

with the

316 ABSOLUTE

ERROR

IN

z

.020 .015 .010 r<

.005 .000

,-4 - . 0 0 5 c~ - . 0 1 0 "~ - . 0 1 5 i

I

I

-3

-2

-1

-.020 --z

0

I

J

I

1

2

3

4

=

Fig. 3. Absolute error in computing x, given qb(x), using eqn. (2).

latter are small for x >1 - 3. The right-hand side member of eqn. (1) is easy to invert to obtain x, given ~ ( x ) : x=~-[8-14(In(0.6931/O(x))],

for O ( x ) ~ < 0 . 5

(2)

.006 Cb

A

.004

,L~ = - - 3

3 a:

r<

I~ = - 1

.002

/2,=_3

2

.000 "~ - . 0 0 2 t~=1

~4

1

~

o

-.004 -.006

i

i 10

.0

i

i 2.0

~

I 3.0

~4

I

-2

4.0

.o

.006 ~4

"~iz=3

#=-1

a~

I

~

I

1.0

I

I

2.0

I

J

3.0

4.0

4

.004

B

.002

~

3

if=-1 / , J

~

~

it=-3 I

/

~ a~ a~

2

~

o

t~ = - 3 =1

B

.000

-.oo2

I~ = 3

-.006

I .0

I 10

I

~ 2.0

i

I 3.0

(1

Fig. 4. Absolute error in / 1 ( - oo, oo; /~, o) (see eqn. (3) for its definition) for several values of/~ and o in two cases: (A) when eqn. (1) is used recursively; and (B) when the approximation given in Ref. [2] is used in the second stage of calculation.

o4 4.0

-2

I .0

L 1.0

I

L 2.0

I

I 3.0

k

I 4.0

(T

Fig. 5. Relative error in 11(- oo, oo; /.t, o) (see eqn. (3) for its definition) for several values of ff and o in two cases: (A) when eqn. (1) is used recursively; and (B) when the approximation given in Ref. [2] is used in the second stage of calculation.

317

The errors on the estimation of x are presented in Fig. 3. Equation (2) can be used to simulate Gaussian deviations with the inverse method. Although the proposed approximation is satisfactory for many applications, a powerful use consists of replacing ~ ( . ) in integrals of the form

,r(a, b; t,,

=

du

(3)

where ~ ( . ) denotes the Gaussian density function with mean ~ and standard deviation o. These integrals, in general, do not possess closed solutions, and replacing d~(u) with approximations based on ratios of polynomials would not lead to a closed form. However, use of eqn. (1) instead of ~ ( . ) in eqn. (3) produces an approximate analytical solution. Integrals of this kind appear, for instance, when computing reliability under uncertain loading, or when calculating expected losses due to earthquake damage [4]. To calibrate the approximation, we compute I 1 ( - o o , oo; #, o ) - - w h i c h can be analytically solved--replacing eqn. (1) in the integrand of eqn. (3). The closed solution requires the evaluation of ~ ( x ) , so there are two possibilities: the use eqn. (1) recursively or to use a more accurate approximation in this second stage. We present in Fig. 4 the absolute errors, for several values of/~ and o, induced by the approximation in both cases: (A) with the recursive use of eqn.

.006 e<

.004 .002 .000 -. 002 -.004

-. 006

,LI,= - - 3

6

~4

c~

'

I

I

1.0

.0

I

I

I

2.0

r¢ c4 r4

4

~

2

~

0

I

3.0

I

-2

4.0

t

I

1.0

.0

f

l

2.0

r

3.0

40

(7

.006

8

e¢

.004

6

~4

.002

a~ ~

#=-3

B

4

.000

l,t=3

2

-. 002

~=1

-.004 I

-.006

I

i]3- = -- 1

I

I

I --2

.0

1.0

2.0

3.0

4.0

(7

Fig. 6. Absolute error in 12(- oo, ¢~; F, o) (see eqn. (3) for its definition) for several values of/~ and o in two cases: (A) when ¢qn. (1) is used recursively; and (B) when the approximation given in Ref. [2] is used in the second stage of calculation.

I

.0

p

1.0

I

i

2.0

i

I

3.0

i

4.0

(7

Fig. 7. Relative error in 12(- o0, oo; ~, a) (see eqn. (3) for its definition) for several values of ~ and o in two cases: (A) when eqn. (1) is used recursively; and (B) when the approximation given in Ref. [2] is used in the second stage of calculation.

318 (1); and (B) with the use of the expressions given in Ref. [2] to compute ~ ( x ) in the resulting function. The corresponding relative errors are depicted in Fig. 5. We also computed ~ 2 ( - ~ , ~ ; ~, a ) - - w h i c h has not an analytical s o l u t i o n - - w i t h eqn. (1), for different values of ~ and o. Results, in terms of absolute and relative errors, are presented in Figs. 6 and 7, where the exact solutions were computed with Simpson's rule and a very small step. The errors appear to be small enough for applications where speed is as valuable as precision.

ACKNOWLEDGEMENTS I am grateful to E. Rosenblueth for his encouragement and critical review of the manuscript. I also appreciate the constructive suggestions made by the reviewers.

REFERENCES 1 J.-T. Lin, Alternatives to Hamaker's approximation to the cumulative normal distribution and its inverse, The Statistician, 37 (1988) 413-414. 2 M. Zelen and N.C. Severo, Probability functions, in: M. Abramowitz and I. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, NY, 1965. 3 E. Rosenblueth, On computing normal refiabilities, Structural Safety, 2 (3) (1985) 165-167. 4 E. Rosenblueth, What should we do with structural reliabilities?, ICASP 5, Vancouver, Canada, May 1987, pp. 25-29.