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1-![](/wiki-images/8/84/Fpc_source_logo.png)
1+[![fpc source logo.png](/wiki-images/e/e1/fpc_source_logo.png)](</File:fpc_source_logo.png>)
22
3-**Liczby losowe** są ważnymi zasobami do zastosowań naukowych, edukacji, tworzenia gier i wizualizacji. Odgrywają kluczową rolę w symulacji numerycznej.
3+**Liczby losowe** są ważnymi zasobami do zastosowań naukowych, edukacji, tworzenia gier i wizualizacji. Odgrywają kluczową rolę w symulacji numerycznej.
44
5-Liczby losowe generowane przez algorytm to liczby pseudolosowe. Należą do (dużego) zbioru powtarzających się liczb, których kolejność jest niemożliwa lub przynajmniej trudna do przewidzenia. W przeciwieństwie do Delphi, który używa liniowego generatora kongruencjalnego (patrz [[Delphi compatible LCG Random|LCG Random kompatybilny z Delphi]]), Free Pascal używa algorytmu MersenneTwister dla swojej standardowej funkcji losowej [[doc:rtl/system/random.html|random]] zdefiniowanej w RTL. Przed pierwszym użyciem generator liczb losowych FPC musi zostać zainicjowany pojedynczym wywołaniem funkcji [[doc:rtl/system/randomize.html|randomize]], która ustawia ziarno generatora. Najlepiej jest to zrobić w fazie uruchamiania programu.
5+Liczby losowe generowane przez algorytm to liczby pseudolosowe. Należą do (dużego) zbioru powtarzających się liczb, których kolejność jest niemożliwa lub przynajmniej trudna do przewidzenia. W przeciwieństwie do Delphi, który używa liniowego generatora kongruencjalnego (patrz [[Delphi compatible LCG Random|LCG Random kompatybilny z Delphi]]), Free Pascal używa algorytmu MersenneTwister dla swojej standardowej funkcji losowej `[random](<http://lazarus-ccr.sourceforge.net/docs/rtl/system/random.html> "doc:rtl/system/random.html")` zdefiniowanej w RTL. Przed pierwszym użyciem generator liczb losowych FPC musi zostać zainicjowany pojedynczym wywołaniem funkcji `[randomize](<http://lazarus-ccr.sourceforge.net/docs/rtl/system/randomize.html> "doc:rtl/system/randomize.html")`, która ustawia ziarno generatora. Najlepiej jest to zrobić w fazie uruchamiania programu.
66
7-Alternatywnie w systemach opartych na systemach Unix i Linux dostępne są urządzenia wirtualne [[Dev random|/dev/random]] i `/dev/urandom`. Generują liczby (pseudo) losowe na podstawie sprzętu.
7+Alternatywnie w systemach opartych na systemach Unix i Linux dostępne są urządzenia wirtualne `[[Dev random|/dev/random]]` i `/dev/urandom`. Generują liczby (pseudo) losowe na podstawie sprzętu.
88
9-Trzecią możliwością jest użycie liczb losowych ze źródeł zewnętrznych, albo ze specjalistycznych urządzeń sprzętowych, albo ze źródeł publicznych, np. na podstawie danych dotyczących rozpadu promieniotwórczego.
9+Trzecią możliwością jest użycie liczb losowych ze źródeł zewnętrznych, albo ze specjalistycznych urządzeń sprzętowych, albo ze źródeł publicznych, np. na podstawie danych dotyczących rozpadu promieniotwórczego.
1010
11+## Contents
12+
13+ * 1 Jednorodny rozkład
14+ * 2 Normal (Gaussian) Distribution
15+ * 3 Exponential Distribution
16+ * 4 Gamma Distribution
17+ * 5 Erlang Distribution
18+ * 6 Poisson Distribution
19+ * 7 t Distribution
20+ * 8 Chi Squared Distribution
21+ * 9 F Distribution
22+ * 10 See also
23+ * 11 References
24+
1125## Jednorodny rozkład
1226
13-Ciągły rozkład jednorodny (nazywany również rozkładem prostokątnym) reprezentuje rodzinę symetrycznych rozkładów prawdopodobieństwa. Tutaj dla każdego członka rodziny wszystkie przedziały tej samej długości na nośniku rozkładu są jednakowo prawdopodobne.
27+Ciągły rozkład jednorodny (nazywany również rozkładem prostokątnym) reprezentuje rodzinę symetrycznych rozkładów prawdopodobieństwa. Tutaj dla każdego członka rodziny wszystkie przedziały tej samej długości na nośniku rozkładu są jednakowo prawdopodobne.
1428
15-Standardowa funkcja [[RTL]] [[doc:rtl/system/random.html|random]] generuje liczby losowe, które mają jednolity rozkład. Jeśli zostanie wywoływana bez parametru zwróci pseudolosową liczbę zmiennoprzecinkową w przedziale \[0, 1), tj. 0 \<= wynik \< 1. Jeśli random jest wywoływana z argumentem longint L, dostarcza losową liczbę zmiennoprzecinkową w przedziale \[0, L).
29+Standardowa funkcja [[RTL]] `[random](<http://lazarus-ccr.sourceforge.net/docs/rtl/system/random.html> "doc:rtl/system/random.html")` generuje liczby losowe, które mają jednolity rozkład. Jeśli zostanie wywoływana bez parametru zwróci pseudolosową liczbę zmiennoprzecinkową w przedziale [[Marsaglia's pseudo random number generators|0, 1), tj. 0 <= wynik < 1\. Jeśli random jest wywoływana z argumentem longint L, dostarcza losową liczbę zmiennoprzecinkową w przedziale [0, L).
1630
17-Kolejny zestaw generatorów liczb losowych o równomiernym rozkładzie jest przedstawiony w [[Marsaglia&|generatorach liczb pseudolosowych Marsaglii]].
31+Kolejny zestaw generatorów liczb losowych o równomiernym rozkładzie jest przedstawiony w `[generatorach liczb pseudolosowych Marsaglii]]`.
1832
19-Jednolicie rozmieszczone liczby losowe nie są przydatne w żadnej aplikacji. Do tworzenia liczb losowych z innych rozkładów potrzebne są specjalne algorytmy:
33+Jednolicie rozmieszczone liczby losowe nie są przydatne w żadnej aplikacji. Do tworzenia liczb losowych z innych rozkładów potrzebne są specjalne algorytmy:
2034
2135## Normal (Gaussian) Distribution
2236
23-One of the more common algorithms to produce normally distributed random numbers from uniformly distributed random numbers is the [Box-Müller approach](http://en.wikipedia.org/wiki/Box–Muller_transform). The following function calculates Gaussian-distributed random numbers:
24-
25-``` pascal
26- function rnorm (mean, sd: real): real;
27- {Calculates Gaussian random numbers according to the Box-Müller approach}
28- var
29- u1, u2: real;
30- begin
31- u1 := random;
32- u2 := random;
33- rnorm := mean * abs(1 + sqrt(-2 * (ln(u1))) * cos(2 * pi * u2) * sd);
34- end;
37+One of the more common algorithms to produce normally distributed random numbers from uniformly distributed random numbers is the [Box-Müller approach](<http://en.wikipedia.org/wiki/Box–Muller_transform>). The following function calculates Gaussian-distributed random numbers:
3538```
36-
37-The same algorithm is used by the [[doc:rtl/math/randg.html|randg]] randg function from the [[RTL]] [[doc:rtl/math/index.html|math]] unit:
39+function rnorm (mean, sd: real): real;
40+ {Calculates Gaussian random numbers according to the Box-Müller approach}
41+ var
42+ u1, u2: real;
43+ begin
44+ u1 := random;
45+ u2 := random;
46+ rnorm := mean * abs(1 + sqrt(-2 * (ln(u1))) * cos(2 * pi * u2) * sd);
47+ end;
48+```
3849
39-``` pascal
50+The same algorithm is used by the [randg](<http://lazarus-ccr.sourceforge.net/docs/rtl/math/randg.html> "doc:rtl/math/randg.html") randg function from the [[RTL]] [math](<http://lazarus-ccr.sourceforge.net/docs/rtl/math/index.html> "doc:rtl/math/index.html") unit:
51+```
4052function randg(mean,stddev: float): float;
4153```
4254
4355## Exponential Distribution
4456
45-An exponential distribution occurs frequently in real-world problems. A classical example is the distribution of waiting times between independent Poisson-random events, e.g. the radioactive decay of nuclei \[Press et al. 1989\].
46-
47-The following function delivers a single real random number out of an exponential distribution. *Rate* is the inverse of the mean, and the constant *RESOLUTION* determines the granularity of generated random numbers.
57+An exponential distribution occurs frequently in real-world problems. A classical example is the distribution of waiting times between independent Poisson-random events, e.g. the radioactive decay of nuclei [Press et al. 1989].
4858
49-``` pascal
59+The following function delivers a single real random number out of an exponential distribution. _Rate_ is the inverse of the mean, and the constant _RESOLUTION_ determines the granularity of generated random numbers.
60+```
5061function randomExp(a, rate: real): real;
51-const
52- RESOLUTION = 1000;
53-var
54- unif: real;
55-begin
56- if rate = 0 then
57- randomExp := NaN
58- else
59- begin
60- repeat
61- unif := random(RESOLUTION) / RESOLUTION;
62- until unif <> 0;
63- randomExp := a - rate * ln(unif);
64- end;
65-end;
62+ const
63+ RESOLUTION = 1000;
64+ var
65+ unif: real;
66+ begin
67+ if rate = 0 then
68+ randomExp := NaN
69+ else
70+ begin
71+ repeat
72+ unif := random(RESOLUTION) / RESOLUTION;
73+ until unif <> 0;
74+ randomExp := a - rate * ln(unif);
75+ end;
76+ end;
6677```
6778
6879## Gamma Distribution
6980
70-The gamma distribution is a two-parameter family of continuous random distributions. It is a generalization of both the exponential distribution and the Erlang distribution. Possible applications of the gamma distribution include modelling and simulation of waiting lines, or queues, and actuarial science.
71-
72-The following function delivers a single real random number out of a gamma distribution. The shape of the distribution is defined by the parameters *a*, *b* and *c*. The function makes use of the function **randomExp** as defined above.
81+The gamma distribution is a two-parameter family of continuous random distributions. It is a generalization of both the exponential distribution and the Erlang distribution. Possible applications of the gamma distribution include modelling and simulation of waiting lines, or queues, and actuarial science.
7382
74-``` pascal
83+The following function delivers a single real random number out of a gamma distribution. The shape of the distribution is defined by the parameters _a_ , _b_ and _c_. The function makes use of the function **randomExp** as defined above.
84+```
7585function randomGamma(a, b, c: real): real;
76-const
77- RESOLUTION = 1000;
78- T = 4.5;
79- D = 1 + ln(T);
80-var
81- unif: real;
82- A2, B2, C2, Q, p, y: real;
83- p1, p2, v, w, z: real;
84- found: boolean;
85-begin
86- A2 := 1 / sqrt(2 * c - 1);
87- B2 := c - ln(4);
88- Q := c + 1 / A2;
89- C2 := 1 + c / exp(1);
90- found := False;
91- if c < 1 then
92- begin
93- repeat
94- repeat
95- unif := random(RESOLUTION) / RESOLUTION;
96- until unif > 0;
97- p := C2 * unif;
98- if p > 1 then
86+ const
87+ RESOLUTION = 1000;
88+ T = 4.5;
89+ D = 1 + ln(T);
90+ var
91+ unif: real;
92+ A2, B2, C2, Q, p, y: real;
93+ p1, p2, v, w, z: real;
94+ found: boolean;
95+ begin
96+ A2 := 1 / sqrt(2 * c - 1);
97+ B2 := c - ln(4);
98+ Q := c + 1 / A2;
99+ C2 := 1 + c / exp(1);
100+ found := False;
101+ if c < 1 then
99102 begin
100103 repeat
101- unif := random(RESOLUTION) / RESOLUTION;
102- until unif > 0;
103- y := -ln((C2 - p) / c);
104- if unif <= power(y, c - 1) then
105- begin
106- randomGamma := a + b * y;
107- found := True;
108- end;
104+ repeat
105+ unif := random(RESOLUTION) / RESOLUTION;
106+ until unif > 0;
107+ p := C2 * unif;
108+ if p > 1 then
109+ begin
110+ repeat
111+ unif := random(RESOLUTION) / RESOLUTION;
112+ until unif > 0;
113+ y := -ln((C2 - p) / c);
114+ if unif <= power(y, c - 1) then
115+ begin
116+ randomGamma := a + b * y;
117+ found := True;
118+ end;
119+ end
120+ else
121+ begin
122+ y := power(p, 1 / c);
123+ if unif <= exp(-y) then
124+ begin
125+ randomGamma := a + b * y;
126+ found := True;
127+ end;
128+ end;
129+ until found;
109130 end
110- else
131+ else if c = 1 then
132+ { Gamma distribution becomes exponential distribution, if c = 1 }
111133 begin
112- y := power(p, 1 / c);
113- if unif <= exp(-y) then
114- begin
115- randomGamma := a + b * y;
116- found := True;
117- end;
118- end;
119- until found;
120- end
121- else if c = 1 then
122- { Gamma distribution becomes exponential distribution, if c = 1 }
123- begin
124- randomGamma := randomExp(a, b);
125- end
126- else
127- begin
128- repeat
129- repeat
130- p1 := random(RESOLUTION) / RESOLUTION;
131- until p1 > 0;
132- repeat
133- p2 := random(RESOLUTION) / RESOLUTION;
134- until p2 > 0;
135- v := A2 * ln(p1 / (1 - p1));
136- y := c * exp(v);
137- z := p1 * p1 * p2;
138- w := B2 + Q * v - y;
139- if (w + D - T * z >= 0) or (w >= ln(z)) then
134+ randomGamma := randomExp(a, b);
135+ end
136+ else
140137 begin
141- randomGamma := a + b * y;
142- found := True;
138+ repeat
139+ repeat
140+ p1 := random(RESOLUTION) / RESOLUTION;
141+ until p1 > 0;
142+ repeat
143+ p2 := random(RESOLUTION) / RESOLUTION;
144+ until p2 > 0;
145+ v := A2 * ln(p1 / (1 - p1));
146+ y := c * exp(v);
147+ z := p1 * p1 * p2;
148+ w := B2 + Q * v - y;
149+ if (w + D - T * z >= 0) or (w >= ln(z)) then
150+ begin
151+ randomGamma := a + b * y;
152+ found := True;
153+ end;
154+ until found;
143155 end;
144- until found;
145- end;
146-end;
156+ end;
147157```
148158
149159## Erlang Distribution
150160
151-The Erlang distribution is a two parameter family of continuous probability distributions. It is a generalization of the exponential distribution and a special case of the gamma distribution, where *c* is an integer. The Erlang distribution has been first described by Agner Krarup Erlang in order to model the time interval between telephone calls. It is used for queuing theory and for simulating waiting lines.
152-
153-``` pascal
154- function randomErlang(mean: real; k: integer): real;
155- const
156- RESOLUTION = 1000;
157- var
158- i: integer;
159- unif, prod: real;
160- begin
161- if (mean <= 0) or (k < 1) then
162- randomErlang := NaN
163- else
164- begin
165- prod := 1;
166- for i := 1 to k do
161+The Erlang distribution is a two parameter family of continuous probability distributions. It is a generalization of the exponential distribution and a special case of the gamma distribution, where _c_ is an integer. The Erlang distribution has been first described by Agner Krarup Erlang in order to model the time interval between telephone calls. It is used for queuing theory and for simulating waiting lines.
162+```
163+function randomErlang(mean: real; k: integer): real;
164+ const
165+ RESOLUTION = 1000;
166+ var
167+ i: integer;
168+ unif, prod: real;
167169 begin
168- repeat
169- unif := random(RESOLUTION) / RESOLUTION;
170- until unif <> 0;
171- prod := prod * unif;
170+ if (mean <= 0) or (k < 1) then
171+ randomErlang := NaN
172+ else
173+ begin
174+ prod := 1;
175+ for i := 1 to k do
176+ begin
177+ repeat
178+ unif := random(RESOLUTION) / RESOLUTION;
179+ until unif <> 0;
180+ prod := prod * unif;
181+ end;
182+ randomErlang := -mean * ln(prod);
183+ end;
172184 end;
173- randomErlang := -mean * ln(prod);
174- end;
175- end;
176185```
177186
178187## Poisson Distribution
179188
180-The Poisson distribution applies to integer values. It represents the probability of *k* successes, when the probability of a success in each trial is small and the rate of occurrence (the mean value) is constant.
181-
182-``` pascal
189+The Poisson distribution applies to integer values. It represents the probability of _k_ successes, when the probability of a success in each trial is small and the rate of occurrence (the mean value) is constant.
190+```
183191function randomPoisson(mean: integer): integer;
184-{ Generator for Poisson distribution (Donald Knuth's algorithm) }
185-const
186- RESOLUTION = 1000;
187-var
188- k: integer;
189- b, l: real;
190-begin
191- assert(mean > 0, 'mean < 1');
192- k := 0;
193- b := 1;
194- l := exp(-mean);
195- while b > l do
196- begin
197- k := k + 1;
198- b := b * random(RESOLUTION) / RESOLUTION;
199- end;
200- randomPoisson := k - 1;
201-end;
192+ { Generator for Poisson distribution (Donald Knuth's algorithm) }
193+ const
194+ RESOLUTION = 1000;
195+ var
196+ k: integer;
197+ b, l: real;
198+ begin
199+ assert(mean > 0, 'mean < 1');
200+ k := 0;
201+ b := 1;
202+ l := exp(-mean);
203+ while b > l do
204+ begin
205+ k := k + 1;
206+ b := b * random(RESOLUTION) / RESOLUTION;
207+ end;
208+ randomPoisson := k - 1;
209+ end;
202210```
203211
204212## t Distribution
205213
206-The t distribution (also referred to a Student's t distribution, since it was published by William Sealy Gosset in 1908 under the pseudonym *Student*) is a continuous probability distribution. Its shape is defined by one parameter, the degrees of freedom (*df*). In statistics, many estimators are t distributed. Therefore, Student's t-distribution plays a major role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The t-distribution also arises in Bayesian analysis of data from a normal family.
207-
208-The following algorithm depends on the [[RTL]] function [[doc:rtl/system/random.html|random]] and on the **randomChisq** function
214+The t distribution (also referred to a Student's t distribution, since it was published by William Sealy Gosset in 1908 under the pseudonym _Student_) is a continuous probability distribution. Its shape is defined by one parameter, the degrees of freedom (_df_). In statistics, many estimators are t distributed. Therefore, Student's t-distribution plays a major role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The t-distribution also arises in Bayesian analysis of data from a normal family.
209215
210-``` pascal
216+The following algorithm depends on the [[RTL]] function `[random](<http://lazarus-ccr.sourceforge.net/docs/rtl/system/random.html> "doc:rtl/system/random.html")` and on the **randomChisq** function
217+```
211218function randomT(df: integer): real;
212-{ Generator for Student's t distribution }
213-begin
214- if df < 1 then
215- randomT := NaN
216- else begin
217- randomT := randg(0, 1) / sqrt(randomChisq(df) / df);
218- end;
219-end;
219+ { Generator for Student's t distribution }
220+ begin
221+ if df < 1 then
222+ randomT := NaN
223+ else begin
224+ randomT := randg(0, 1) / sqrt(randomChisq(df) / df);
225+ end;
226+ end;
220227```
221228
222229## Chi Squared Distribution
223230
224-The chi squared distribution is a continuous distribution of random numbers with *df* degrees of freedom. It is the distribution of a sum of the squares of *df* independent standard normal random variables. The chi squared distribution has numerous applications in inferential statistics, e.g. in estimating variances and for chi-squared tests. It is a special gamma distribution with *c* = *df*/ 2 and *b* = 2. Therefore the following function depends on the function **randomGamma**.
225-
226-``` pascal
231+The chi squared distribution is a continuous distribution of random numbers with _df_ degrees of freedom. It is the distribution of a sum of the squares of _df_ independent standard normal random variables. The chi squared distribution has numerous applications in inferential statistics, e.g. in estimating variances and for chi-squared tests. It is a special gamma distribution with _c_ = _df_ / 2 and _b_ = 2. Therefore the following function depends on the function **randomGamma**.
232+```
227233function randomChisq(df: integer): real;
228-begin
229- if df < 1 then
230- randomChisq := NaN
231- else
232- randomChisq := randomGamma(0, 2, 0.5 * df);
233-end;
234+ begin
235+ if df < 1 then
236+ randomChisq := NaN
237+ else
238+ randomChisq := randomGamma(0, 2, 0.5 * df);
239+ end;
234240```
235241
236242## F Distribution
237243
238-The F distribution, also referred to as Fisher-Snedecor distribution, is a continuous probability distribution. It is used for F Test and ANOVA. It has two degrees of freedom that serve as shape parameters *v* and *w* and that are positive integers. The following function **randomF** makes use of **randomChisq**.
239-
240-``` pascal
244+The F distribution, also referred to as Fisher-Snedecor distribution, is a continuous probability distribution. It is used for F Test and ANOVA. It has two degrees of freedom that serve as shape parameters _v_ and _w_ and that are positive integers. The following function **randomF** makes use of **randomChisq**.
245+```
241246function randomF(v, w: integer): real;
242-begin
243- if (v < 1) or (w < 1) then
244- randomF := NaN
245- else
246- randomF := randomChisq(v) / v / (randomChisq(w) / w);
247-end;
247+ begin
248+ if (v < 1) or (w < 1) then
249+ randomF := NaN
250+ else
251+ randomF := randomChisq(v) / v / (randomChisq(w) / w);
252+ end;
248253```
249254
250255## See also
251256
252-- [[Dev random]]
253-- [[Functions for descriptive statistics]]
254-- [[Marsaglia&|Marsaglia's pseudo random number generators]]
255-- [[A simple implementation of the Mersenne twister]]
256-- [[Delphi compatible LCG Random]]
257-- [[EverettRandom]]
257+ * [[Dev random]]
258+ * [[Functions for descriptive statistics]]
259+ * [[Marsaglia's pseudo random number generators]]
260+ * [[A simple implementation of the Mersenne twister]]
261+ * [[Delphi compatible LCG Random]]
262+ * [[EverettRandom]]
258263
259264## References
260265
261-1. [G. E. P. Box and Mervin E. Muller, *A Note on the Generation of Random Normal Deviates*, The Annals of Mathematical Statistics (1958), Vol. 29, No. 2 pp. 610–611](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177706645)
262-2. Dietrich, J. W. (2002). [Der Hypophysen-Schilddrüsen-Regelkreis](http://openlibrary.org/books/OL24586469M/Der_Hypophysen-Schilddrüsen-Regelkreis). Berlin, Germany: Logos-Verlag Berlin. ISBN 978-3-89722-850-4. OCLC 50451543.
263-3. Press, W. H., B. P. Flannery, S. A. Teukolsky, W. T. Vetterling (1989). [Numerical Recipes in Pascal](http://openlibrary.org/works/OL16807779W/). The Art of Scientific Computing, Cambridge University Press, ISBN 0-521-37516-9.
264-4. Richard Saucier, [Computer Generation of Statistical Distributions](http://ftp.arl.mil/random/random.pdf), ARL-TR-2168, US Army Research Laboratory, Aberdeen Proving Ground, MD, 21005-5068, March 2000.
265-5. R.U. Seydel, Generating Random Numbers with Specified Distributions. In: Tools for Computational Finance, Universitext, [DOI 10.1007/978-1-4471-2993-6_2](http://dx.doi.org/10.1007/978-1-4471-2993-6_2), © Springer-Verlag London Limited 2012
266-6. Christian Walck, Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists, Internal Report SUF–PFY/96–01, University of Stockholm 2007
266+ 1. [G. E. P. Box and Mervin E. Muller, _A Note on the Generation of Random Normal Deviates_ , The Annals of Mathematical Statistics (1958), Vol. 29, No. 2 pp. 610–611](<http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aoms/1177706645>)
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269+ 4. Richard Saucier, [Computer Generation of Statistical Distributions](<http://ftp.arl.mil/random/random.pdf>), ARL-TR-2168, US Army Research Laboratory, Aberdeen Proving Ground, MD, 21005-5068, March 2000.
270+ 5. R.U. Seydel, Generating Random Numbers with Specified Distributions. In: Tools for Computational Finance, Universitext, [DOI 10.1007/978-1-4471-2993-6_2](<http://dx.doi.org/10.1007/978-1-4471-2993-6_2>), © Springer-Verlag London Limited 2012
271+ 6. Christian Walck, Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists, Internal Report SUF–PFY/96–01, University of Stockholm 2007
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269-*From the Free Pascal wiki, page "Generating Random Numbers/pl", as it stood on 2020-11-08 after a change by Slawek, out of the wiki's dump of 2024-05-26. Copyright stays with the author who wrote it, according to the old wiki's [[Licensing]] page. If you have concerns about this material, [message the admin](/wiki-contact).*
274+*From the Free Pascal wiki, page "Generating Random Numbers/pl", as the Internet Archive kept it on 2022-08-09. Copyright stays with the author who wrote it, according to the old wiki's [[Licensing]] page. If you have concerns about this material, [message the admin](/wiki-contact).*

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