Numerical method for calculating cumulative distribution of positive random variabl

Helstrom, C. W.

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Approximate Calculation  of Cumulative Probability
from a Moment -Generating Function 
Carl W. Helstrom 
Department of Applied Electrophysics 
University of California, San Diego 
La Jolla, California 92037 
ABSTRACT 
A numerical method is presented for calcu-
lating the cumulative distribution of a positive 
random variable from its moment-generating function. 
It involves an expansion of the rectangular function 
in Laguerre functions. As examples, the cumulative 
exponential and cumulative Poisson probability func-
tions are approximated.
1
https://ntrs.nasa.gov/search.jsp?R=19690006420 2020-03-12T08:18:25+00:00Z
I 
A common problem is the caicuiJtion of the cumulative probability 
distribution	 x 
Q(x)	 p(y) dy, 0 < x	 (1) 
• of a positive random variable y of which one knows only the moment-
generating function (m. g. f.),
Co 
(s) _E(eYS) =$ eYS p(y) dy,	 (2) 
where p(y) is the probability density function (p. d. f.) of y. 
- In signal detection theory, for instance, y is related to the likelihood 
ratio, and I-Q(x) is the false-alarm or detection probability for a decision 
level x. Often the m. g. f. can be worked out rather easily, but it is 
	
•	 impossible to determine p(y) from p (s) analytically by, for instance, taking 
the inverse Laplace transform of (-s) or the inverse Fourier transform of 
A technique for calculating Q(x) numerically can be derived by writing 
(2)as
Q 	 = $ R(y/x)p (y) dy	 (3) 
where R(t) is the rectangular function 
R(t) = 1, 0 <t < 1; R(t) = 0, t >1. 	 (4) 
•	 One expands R(t) in a series of Laguerre functions, 
Co 
	
•	
• R(t) =e_kt/2	 am Lm(kt)	 (5) 
mO 
2 
•	 where
a = k I l _kt/2 e 	 L (k t) dt =' M.	 m	 • 
2e	 FL	 (k) - L (k) l - a	 (6) m	 m-1 
	
•	
- 1-
The series in (5) is to be truncated at a finite number M of terms. 
The cumulative distribution is 
Q(x)=a 
m m 
C (x),	 7 
M=O 
where the coefficients
CO 
Cm(X) = $ e_ky/2x Lm(kY/X) p(y)dy	 (8) 
can be expressed in terms of ji (-k/2x) and its derivatives. In particular, 
C 0 (x)=(-k/2x),	 (9) 
and by using the fomula 3 
L(t) =(-l)mtm/m! -	 (i)r() Lmr(t)
	
(10) 
a recurrence relation for C m (x) is easily obtained. 
m 
•	
Cm(X) = 2m(m!)1{sm :
	 [ (s)] s = -k/Zx
	
()r (I) C(x),	 (1 1) 
which facilitates numerical computation. 
The method was tried out with two very different distributions, 
the exponential,
p(y) =e, y > 0; p(y) = 0, y < 0,	 (12) 
whose m. g. f. is 
•	 (s) = (s_i) l, Rt s < 1,	 (13) 
and the Poisson,	 •CO 
p(y) = e	 ô(y-n)/n!	 (14) 
whose m.g.f. is	 • • 
(s)	 èxp [ X (e 5 - 1)]	 (15) 
	
Firstit was necessary to determine the best value of the scale para- • 	 • 
meter k when M terms are used. This was done by hunting the value of k 	 • 
-2-
that yielded the minimum mean-square error 
M - l 
e = i -;
	
a m 2
	 (16) 
in fitting the truncated version of (5) to R(t) For M20, we found that 
k z43 gives a mean-square error 	 = 0.01567. The coefficients am are
listed in Table 1. 
For the exponential p. d. f. we list in Table 2 the percentage error in 
Q(x) for 0 < Q(x) < 1/2 and the percentage error in 1 - Q(x) for 1/2 < Q(x) < I. 
The relative error decreases with increasing x. 
For the Poissondistribution we evaluated Q(x) by the approximation 
method for values of x halfway between the integers and compared the results 
with the Poisson distribution summed from y = 0 to the greatest integer in x. 
Table 3 lists the percentage errors in Q(x) for 0 < Q(x) < 1/2 and in 1 - Q(x) 
for 1/2 < Q(x) < I. 
The accuracy is greatest near the mean and poorest in the tails of 
the Poisson distribution, and this can be expected in most applications. 
There exist other approximation methods best suited for the tails of a dis-
tribution. For large x the inverse Laplace transform of i (-s) can be 
approximated by the method of steepest descents. For x near 0, an 
approximation to Q(x) can be obtained from the asymptotic behavior  of 
L (-s) for large s. The method described here fills the gap. 
An alternative method is the Edgeworth series, but it has an 
asymptotic character that restricts its usefulness. 6 There is an optimum 
number of terms in the Edgeworth series, and if more are used, the 
ISM
accuracy decreases markedly. Numerical Fourier transformation of p (i w ), 
followed by numerical integration of the p. d. f. p(y), might be used in some 
cases, but would hardly be suitable for a discrete random variable like the 
Poisson-distributed one of our second example. 
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p.
TABLE 1 
Coefficients of Lague rre Expansion 
m	 0	 1	 2	 3 4 5 6 
a	 2	 -2	 2	 -2 1.9999 -1.9992 1.9956 
ni 
m	 9 10 11 12 
a	 -1.7766 1.4460 -0.87710 0.15695 
m 
m	 15 16 17 18 
a	 0.071648 0. 33527 -0. 22910 -0. 18569 
m
7	 8 
-1.9800	 1.9261 
13	 14 
0.41416 -0.49039 
19 
0.24351 
X	 0.1 
Error (07°) 0. 743 
X
	 3.0 
Error (%) -0. 170
TABLE 2 
Exponential Distribution 
0.3 0.5 1.0 
0.561 0.422 -0.351 
4.0 5.0 6.0 
-0.0878 -0.0408 -0.0174
1.5 2.0 
-0.336 -0.285 
8.0 10.0 
-0.00239 -0.000224 
TABLE 3 
Poisson Distribution 
X 6 8 10 12 14 16 
Q 0. 00763 0.0374 0. 1185 0.2676 0,4656 o.6641 
Error (%) -59.6 -17. 6 -2.82 -0. 133 0. 00493 0. 135 
X 18 20 22 24 26 28	 30... 
Q 0.8195 0. 9170 0. 9673 0. 9888 0. 99669 0. 99914	 0.99980 
Error (%) 0.241 0. 735 1. 199 1. 379 -0.415 -20.2	 72.2
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FOOTNOTES 
This research was supported by Grant #05-009-079 of the National 
Aeronautics and Space Administration. 
1. J. W. Head and W. P. Wilson, "Laguerre Functions: Tables and 
properties, Proceedings of the Institute of Electrical Engineers, 
vol. 103C, pp. 428-436; June, 1956. 
2. See eq. (38), reference 1, p. 434. 
3. See eq. (67), reference 1, p. 435. 
4. G. Doetsch, Handbuch der Laplace -Transformation, Birkhuser 
Verlag, Basel and Stuttgart, 1955, vol. 2, ch. 3, § 5, pp. 83-88. 
5. Ibid, vol. 2, ch. 3, § 1, pp. 45-50 and § 7, pp. 92-94. 
6. T. C. Fry, Probability and Its Engineering Uses, D. Van Nostrand 
Co., Princeton, 2nd ed., 1965, p. 262. 
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Approximate calculation of cumulative probability from a moment-generating function

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