\input style
S IZBYTKOM KOMPENSIRUET |TOT NEDOSTATOK. pROGRAMMU DLYA METODA mARSALXI
NAPISATX GORAZDO TRUDNEE, NO ESLI PODPROGRAMMU, OSNOVANNUYU NA ALGORITME~M,
SOSTAVITX V OBSHCHEM VIDE, ONA YAVITSYA CENNYM VKLADOM V LYUBUYU BIBLIOTEKU
PODPROGRAMM. mNOGOCHISLENNYE PRIMENENIYA NORMALXNO RASPREDELENNYH
SLUCHAJNYH VELICHIN TREBUYUT BOLXSHOGO KOLICHESTVA SLUCHAJNYH CHISEL, TAK
CHTO VAZHNA SKOROSTX IH VYRABOTKI.

dOPOLNITELXNUYU INFORMACIYU O METODE tEJCHROEVA, A TAKZHE OBZOR NEKOTORYH
DRUGIH METODOV, HUDSHIH, KAK TEPERX VYYASNILOSX, CHEM OBSUZHDAEMYE ZDESX,
MOZHNO POLUCHITX IZ STATXI m.~mALLERA ({\sl JACM,\/} {\bf 6} (1959), 376--383).

{\sl (5)~rAZNOVIDNOSTI NORMALXNOGO RASPREDELENIYA.\/} mY
RASSMOTRELI NORMALXNOE RASPREDELENIE S NULEVYM SREDNIM ZNACHENIEM I
STANDARTNYM OTKLONENIEM, RAVNYM EDINICE. eSLI $X$~IMEET TAKOE
RASPREDELENIE, TO U FUNKCII RASPREDELENIYA SLUCHAJNOJ VELICHINY
$$
Y=\mu+\sigma X
\eqno(24)
$$
SREDNEE ZNACHENIE RAVNO~$\mu$, A STANDARTNOE OTKLONENIE~$\sigma$.
bOLEE TOGO, ESLI~$X_1$ I~$X_2$---NEZAVISIMYE NORMALXNYE SLUCHAJNYE VELICHINY
SO SREDNIM ZNACHENIEM NULX I EDINICHNYM STANDARTNYM OTKLONENIEM I ESLI
$$
Y_1=\mu_1+\sigma_1 X_1, \qquad
Y_2=\mu_2+\sigma_2(\rho X_1+\sqrt{1-\rho^2}X_2),
\eqno(25)
$$
TO~$Y_1$ I~$Y_2$---\emph{ZAVISIMYE} SLUCHAJNYE VELICHINY, RASPREDELENNYE  SO
SREDNIMI ZNACHENIYAMI~$\mu_1$, $\mu_2$, STANDARTNYMI OTKLONENIYAMI~$\sigma_1$,
$\sigma_2$ I KO|FFICIENTOM KORRELYACII~$\rho$. (oBOBSHCHENIE NA SLUCHAJ~$n$
PEREMENNYH SM.~V~UPR.~13.)

\section{D.~eKSPONENCIALXNOE RASPREDELENIE}. dRUGOJ VAZHNYJ VID SLUCHAJNYH
VELICHIN---VELICHINY S \emph{|KSPONENCIALXNYM RASPREDELENIEM.} tAKIE SLUCHAJNYE
VELICHINY BYVAYUT NUZHNY V ZADACHAH, GDE RASSMATRIVAETSYA "VREMYA POYAVLENIYA".
nAPRIMER, ESLI RADIOAKTIVNOE VESHCHESTVO IZLUCHAET V SREDNEM KAZHDYE
$\mu$~SEKUND ODNU ALXFA-CHASTICU, TO PROMEZHUTKI VREMENI MEZHDU DVUMYA
POSLEDOVATELXNYMI VYLETAMI CHASTIC IMEYUT |KSPONENCIALXNOE RASPREDELENIE SO
SREDNIM ZNACHENIEM~$\mu$. eTO RASPREDELENIE OPREDELYAETSYA FORMULOJ
$$
F(x)=1-e^{-x/\mu}, \rem{$x\ge0$.}
\eqno(26)
$$
oTSYUDA SLEDUET, CHTO ESLI $X$~IMEET |KSPONENCIALXNOE RASPREDELENIE SO
SREDNIM ZNACHENIEM~$1$, TO $\mu X$~PODCHINYAETSYA |KSPONENCIALXNOMU
RASPREDELENIYU SO SREDNIM~$\mu$. pO|TOMU DOSTATOCHNO RASSMOTRETX
SLUCHAJ~$\mu=1$. oBYCHNO ISPOLXZUYUTSYA TRI METODA.

{\sl (1)~lOGARIFMICHESKIJ METOD.\/} yaSNO, CHTO~$y=F(x)=1-e^{-x}$
MOZHNO PREDSTAVITX V VIDE~$x=F^{-1}(y)=-\ln(1-y)$. pO|TOMU, VSLEDSTVIE
%% 142
SOOTNOSHENIYA~(6), VELICHINA~$-\ln(1-U)$ IMEET |KSPONENCIALXNOE RASPREDELENIE.
tAK KAK $1-U$~RASPREDELENA RAVNOMERNO, ESLI~$U$---RAVNOMERNO RASPREDELENNOE
SLUCHAJNOE CHISLO, TO SLUCHAJNAYA VELICHINA
$$
X=-\ln U
\eqno(27)
$$
RASPREDELENA |KSPONENCIALXNO SO SREDNIM ZNACHENIEM, RAVNYM EDINICE. (v
PROGRAMMAH SLEDUET IZBEGATX SLUCHAYA~$U=0$.)

{\sl (2)~mETOD SLUCHAJNOJ MINIMIZACII.\/} sLEDUYUSHCHIJ ALGORITM (dZH.~mARSALXYA)
VYCHISLYAET ZNACHENIYA |KSPONENCIALXNO RASPREDELENNOJ SLUCHAJNOJ VELICHINY BEZ
ISPOLXZOVANIYA PODPROGRAMMY LOGARIFMA.

\alg E.(eKSPONENCIALXNOE RASPREDELENIE SO SREDNIM~$1$.) iSPOLXZUYUTSYA
TABLICY KONSTANT~$P[j]$, $Q[j]$ DLYA~$j\ge 1$, OPREDELENNYE FORMULAMI
$$
P[j]=1-{1\over e^j}, \quad
Q[j]={1\over e-1}\left({1\over1!}+{1\over2!}+\cdots+{1\over j!}\right).
\eqno(28)
$$

dLINA TABLIC OGRANICHIVAETSYA ZNACHENIEM MAKSIMALXNOJ DROBI, KOTORUYU MOZHNO
RAZMESTITX V MASHINNOM SLOVE.

\st[nACHALO DROBNOJ CHASTI.] uSTANOVITX~$j\asg1$. vYRABOTATX SLUCHAJNYE
CHISLA~$U_0$ I~$U_1$ I USTANOVITX~$X\asg -U_1$.

\st[mINIMIZACIYA ZAKONCHENA?] eSLI~$U_0<Q[j]$, PEREJTI K~\stp{4}.

\st[mINIMIZIROVATX.] uSTANOVITX~$j\asg j+1$. vYRABOTATX SLUCHAJNOE
CHISLO~$U_j$; ESLI~$X>U_j$, USTANOVITX~$X\asg U_j$. vERNUTXSYA OBRATNO K SHAGU~\stp{2}.

\st[nACHALO CELOJ CHASTI.] (mY UZHE VYCHISLILI DROBNUYU CHASTX OKONCHATELXNOGO
REZULXTATA, $X$, I DOLZHNY DOBAVITX K NEMU SOOTVETSTVUYUSHCHEE CELOE CHISLO,
CHTOBY ZAKONCHITX VYCHISLENIYA.) vYRABOTATX NOVOE SLUCHAJNOE CHISLO~$U$ I
USTANOVITX~$j\asg 1$.

\st[sDELANA LI POPRAVKA?] eSLI~$U<P[j]$, ALGORITM ZAKANCHIVAETSYA.

\st[pOPRAVKA NA~1.] uSTANOVITX~$j\asg j+1$, $X\asg X+1$ I VERNUTXSYA K SHAGU~\stp{5}.
\algend

chTOBY POKAZATX SPRAVEDLIVOSTX METODA, PROANALIZIRUEM RASPREDELENIE~$X$ V
NACHALE SHAGA~E4. eSLI~$n$---OKONCHATELXNOE ZNACHENIE~$j$, MY
IMEEM~$X=\min(U_1, U_2,~\ldots, U_n)$, GDE~$U_1$, $U_2$,~\dots,
$U_n$---NEZAVISIMYE SLUCHAJNYE CHISLA; PO|TOMU VEROYATNOSTX TOGO, CHTO~$X\le x$,
ESTX~$p_n(x)=1-(1-x)^n$. vEROYATNOSTX TOGO, CHTO~$n$---OKONCHATELXNOE
ZNACHENIE, RAVNA~$Q[n]-Q[n-1]=1/(e-1)n!$. pO|TOMU POLNAYA VEROYATNOSTX
SOBYTIYA~$X\le x$ RAVNA
$$
\sum_{n\ge 1} {p(x)\over (e-1)n!}={e\over e-1}(1-e^{-x}), \rem{$0\le x\le 1$.}
$$
%% 143
aNALOGICHNO, PRI RASSMOTRENII SHAGOV~E4--E6 MY NAHODIM, CHTO~$\floor{X}\le m$
S VEROYATNOSTXYU~$P[m+1]$. oKONCHATELXNO, VEROYATNOSTX TOGO, CHTO~$m\le X \le m+x$, ESTX
$$
(P[m+1]-P[m])\left({e\over e-1}(1-e^{-x}\right)=e^{-m}-e^{-(m+x)},
\rem{$0\le x \le 1$.}
$$
eTO DOKAZYVAET, CHTO $X$~IMEET RASPREDELENIE~$F(x)=1-e^{-x}$ DLYA~$0\le x<\infty$.

eTOT ALGORITM DOVOLXNO BYSTRYJ. v SREDNEM SHAGI~E2 I~E5 VYPOLNYAYUTSYA TOLXKO
$1.582$~RAZ I TOLXKO $2.582$~SLUCHAJNYH CHISLA VYCHISLYAETSYA DLYA POLUCHENIYA
ODNOJ |KSPONENCIALXNOJ SLUCHAJNOJ VELICHINY. eTO MOZHET OKAZATXSYA ZNACHITELXNO
BYSTREE, CHEM VYCHISLENIE LOGARIFMA ODNOGO SLUCHAJNOGO CHISLA. iNTERESNAYA
MODIFIKACIYA METODA SODERZHITSYA V RABOTE
[m.~Sibuya, {\sl Ann.\ Inst.\ Stat.\ Math.,\/} {\bf 13} (1962), 231--237].

{\sl (3)~mETOD PRYAMOUGOLXNIKA---KLINA---HVOSTA.\/} dLYA
NORMALXNOGO RASPREDELENIYA (KAK I DLYA MNOGIH DRUGIH) SUSHCHESTVUET OCHENX
BYSTRYJ METOD, OSNOVANNYJ NA RAZLOZHENII FUNKCII PLOTNOSTI VEROYATNOSTI.
dETALI OBSUZHDAYUTSYA V {\sl CACM,\/} {\bf 7} (1964), 298--300.

\section{E.~dRUGIE NEPRERYVNYE RASPREDELENIYA}. mY PERECHISLIM ZDESX KOROTKO
NEKOTORYE DRUGIE RASPREDELENIYA, KOTORYE BYVAYUT DOVOLXNO CHASTO NUZHNY NA
PRAKTIKE, I METODY MODELIROVANIYA SOOTVETSTVUYUSHCHIH SLUCHAJNYH VELICHIN S
POMOSHCHXYU UZHE IZVESTNYH NAM PRIEMOV.

{\sl $\chi^2$-RASPREDELENIE\/} S $\nu$~STEPENYAMI SVOBODY, KOTOROE TAKZHE
NAZYVAYUT GAMMA-RASPREDELENIEM PORYADKA~$\nu/2$. mY IMEEM
$$
F(x)={1\over 2^{\nu/2} \Gamma(\nu/2)} \int_0^x t^{\nu/2-1}e^{-t/2}\,dt, \rem{$x\ge0$}.
\eqno(29)
$$
eSLI~$\nu=2k$, GDE~$k$---CELOE CHISLO, POLOZHIM~$X=2(Y_1+Y_2+\cdots+Y_k)$,
GDE~$Y_i$---NEZAVISIMYE |KSPONENCIALXNO RASPREDELENNYE SLUCHAJNYE VELICHINY,
SO SREDNIM ZNACHENIEM~$1$ KAZHDAYA. eSLI~$\nu=2k+1$,
POLOZHIM~$X=2(Y_1+\cdots+Y_k)+Z^2$, GDE~$Y_i$---TE ZHE VELICHINY,
a~$Z$---NEZAVISIMAYA NORMALXNO RASPREDELENNAYA SLUCHAJNAYA VELICHINA
(SREDNEE~$0$, DISPERSIYA~$1$). dLYA DOKAZATELXSTVA SM.~UPR.~16. zAMETIM, CHTO,
ESLI~$Y_1$,~\dots, $Y_k$ NAHODYATSYA LOGARIFMICHESKIM METODOM, DLYA
OPREDELENIYA~$Y_1+\cdots+Y_k=-\ln(U_1\ldots{}U_k)$ TREBUETSYA VYCHISLITX
TOLXKO ODIN LOGARIFM.

{\sl (2)~bETA-RASPREDELENIE\/} S~$\nu_1$ I~$\nu_2$ STEPENYAMI SVOBODY,
OPREDELYAETSYA FORMULOJ:
$$
F(x)={\Gamma((\nu_1+\nu_2)/2)\over \Gamma(\nu_1/2)\Gamma(\nu_2/2)}
\int_0^x t^{\nu_1/2-1}(1-t)^{\nu_2/2-1}\,dt, \rem{$0\le x \le 1$.}
\eqno(30)
$$
%% 144
pUSTX~$Y_1$ I~$Y_2$ NEZAVISIMY I IMEYUT $\chi^2$-RASPREDELENIE S~$\nu_1$,
$\nu_2$~STEPENYAMI SVOBODY SOOTVETSTVENNO. pOLAGAEM~$X\asg Y_1/(Y_1+Y_2)$.

dRUGOJ METOD, KOTORYJ OSTAETSYA SPRAVEDLIVYM I DLYA NECELYH~$\nu_1$
I~$\nu_2$, SVODITSYA K VYCHISLENIYU~$Y_1\asg U_1^{2/\nu_1}$,
$Y_2\asg U_2^{2/\nu_2}$ S POVTORENIEM V SLUCHAE NEOBHODIMOSTI |TOGO PROCESSA DO
TEH POR, POKA NE BUDET VYPOLNENO NERAVENSTVO~$Y_1+Y_2\le 1$. nAKONEC,
$X\asg Y_1/(Y_1+Y_2)$. [sM.~M.~D.~J\"ohnk, {\sl Metrika,\/} {\bf 8} (1964), 5--15.]
pO-DRUGOMU, ESLI~$\nu_1=2k_1$, a~$\nu_2=2k_2$---OBA CHETNYE CELYE
CHISLA, MY MOZHEM OPREDELITX~$X$ KAK $k_1\hbox{-E}$ PO PORYADKU NAIMENXSHEE IZ
$k_1+k_2-1$~NEZAVISIMYH SLUCHAJNYH CHISEL.

{\sl (3)~$F$-RASPREDELENIE\/} (RASPREDELENIE OTNOSHENIJ DISPERSIJ)
S~$\nu_1$ I~$\nu_2$~STEPENYAMI SVOBODY OPREDELYAETSYA KAK
$$
F(x)={\nu_1^{\nu_1/2}\nu_2^{\nu_2/2}\Gamma((\nu_1+\nu_2)/2) \over \Gamma(\nu_1/2)\Gamma(\nu_2/2)}
\int_0^x t^{\nu_1/2-1}(\nu_2+\nu_1t)^{-\nu_1/2-\nu_2/2}\,dt, \rem{$x\ge0$.}
\eqno(31)
$$

pUSTX~$Y_1$ I~$Y_2$---NEZAVISIMYE SLUCHAJNYE VELICHINY,
IMEYUSHCHIE $\chi^2$-RASPREDELENIE S~$\nu_1$, $\nu_2$~STEPENYAMI SVOBODY
SOOTVETSTVENNO. pOLOZHIM~$X\asg Y_1\nu_2/Y_2\nu_1$,
ILI~$X\asg\nu_2 Y/\nu_1(1-Y)$, GDE $Y$~IMEET BETA-RASPREDELENIE~(30).

{\sl(4) $t$-RASPREDELENIE\/} S~$\nu$~STEPENYAMI SVOBODY OPREDELYAETSYA KAK
$$
F(x)={\Gamma((\nu+1)/2)\over \sqrt{\pi\nu}\Gamma(\nu/2)} \int_{-\infty}^x (1+t^2/\nu)^{-(\nu+1)/2}\,dt.
\eqno(32)
$$
pUSTX~$Y_1$---NORMALXNO RASPREDELENNAYA SLUCHAJNAYA VELICHINA (SREDNEE~$0$,
DISPERSIYA~$1$), PUSTX~$Y_2$---NEZAVISIMAYA OT~$Y_1$ SLUCHAJNAYA VELICHINA,
IMEYUSHCHAYA $\chi^2$-RASPREDELENIE S $\nu$~STEPENYAMI SVOBODY. pOLOZHIM~$X\asg Y_1/\sqrt{Y_2/\nu}$.

{\sl (5)~sLUCHAJNAYA TOCHKA NA $n$-MERNOJ SFERE S EDINICHNYM
RADIUSOM.\/} pUSTX~$X_1$, $X_2$,~\dots, $X_n$---NEZAVISIMYE NORMALXNYE
SLUCHAJNYE VELICHINY (SREDNEE~$0$, DISPERSIYA~$1$). sLUCHAJNAYA TOCHKA NA
EDINICHNOJ SFERE IMEET KOORDINATY
$$
(X_1/r, X_2/r, \ldots, X_n/r), \rem{GDE $r=\sqrt{X_1^2+X_2^2+\cdots+X_n^2}$.}
\eqno (33)
$$
zAMETIM, CHTO ESLI $X_i$~VYCHISLYAYUTSYA S POMOSHCHXYU METODA POLYARNYH KOORDINAT
(ALGORITM~P), TO KAZHDYJ RAZ V REZULXTATE MY IMEEM DVE NEZAVISIMYE
VELICHINY~$X_1$ I~$X_2$ I~$X_1^2+X_2^2=-2\ln S$ (V OBOZNACHENIYAH
ALGORITMA~P). eTO |KONOMIT NEMNOGO VREMENI, NEOBHODIMOGO DLYA
VYCHISLENIYA~$r$. sPRAVEDLIVOSTX METODA PROISTEKAET IZ TOGO FAKTA, CHTO
FUNKCIYA RASPREDELENIYA TOCHKI~$(X_1,~\dots, X_n)$ IMEET PLOTNOSTX, ZAVISYASHCHUYU
TOLXKO OT RASSTOYANIYA DO CENTRA.
%% 145
pO|TOMU PROEKCIYA TAKOJ TOCHKI NA EDINICHNUYU SFERU IMEET RAVNOMERNOE
RASPREDELENIE. mETOD VPERVYE BYL PREDLOZHEN dZH.~bRAUNOM.

\section{F.~vAZHNYE DISKRETNYE RASPREDELENIYA}. vOOBSHCHE GOVORYA, K
VEROYATNOSTNYM RASPREDELENIYAM SLUCHAJNYH VELICHIN, PRINIMAYUSHCHIH
TOLXKO CELOCHISLENNYE ZNACHENIYA, PRIMENIMY PRIEMY, OPISANNYE V NACHALE |TOGO
RAZDELA. nO NEKOTORYE IZ |TIH RASPREDELENIJ STOLX VAZHNY DLYA PRAKTICHESKIH
PRIMENENIJ, CHTO ZASLUZHIVAYUT SPECIALXNOGO RASSMOTRENIYA.

{\sl (1)~gEOMETRICHESKOE RASPREDELENIE.\/} eSLI NEKOTOROE SOBYTIE
PROISHODIT S VEROYATNOSTXYU~$p$, CHISLO~$N$ NEZAVISIMYH ISPYTANIJ,
NEOBHODIMYH, CHTOBY |TO SOBYTIE PROIZOSHLO (ILI CHISLO ISPYTANIJ MEZHDU
SOBYTIYAMI), PODCHINYAETSYA GEOMETRICHESKOMU RASPREDELENIYU. mY IMEEM~$N=1$ S
VEROYATNOSTXYU~$p$, $N=2$ S VEROYATNOSTXYU~$(1-p)p$,~\dots, $N=n$ S
VEROYATNOSTXYU~$(1-p)^{n-1}p$. (v SUSHCHNOSTI |TO TA ZHE SITUACIYA, S KOTOROJ MY
VSTRECHALISX PRI "PROVERKE INTERVALOV" V~P.~3.3.2; ONA VOZNIKAET, KOGDA MY
INTERESUEMSYA, SKOLXKO RAZ VYPOLNYAYUTSYA OPREDELENNYE CIKLY V ALGORITMAH
|TOGO RAZDELA, NAPRIMER SHAGI~P1--P3 V METODE POLYARNYH KOORDINAT.)

dLYA VYRABOTKI ZNACHENIYA SLUCHAJNOJ VELICHINY S TAKIM RASPREDELENIEM, KOGDA
$p$~MALO, ESTX UDOBNAYA FORMULA:
$$
N=\ceil{\ln U/\ln (1-p)}.
\eqno (34)
$$
chTOBY PROVERITX EE, UBEDIMSYA, CHTO~$\ceil{\ln U/\ln(1-p)}=n$ TOGDA I TOLXKO
TOGDA, KOGDA~$n-1<\ln U/\ln (1-p)\le n$, T.~E.~$(1-p)^{n-1}>U\ge (1-p)^n$,
A |TO PROISHODIT S VEROYATNOSTXYU~$p(1-p)^{n-1}$, CHTO I TREBOVALOSX POKAZATX.

chASTNYJ SLUCHAJ~$p=1/2$ ESHCHE LEGCHE MODELIROVATX NA DVOICHNOJ MASHINE, TAK KAK
FORMULA~(34) PREVRASHCHAETSYA V~$N=\ceil{-\log_2 U}$, T.~E.~$N$ NA EDINICU
BOLXSHE, CHEM CHISLO PERVYH NULEVYH RAZRYADOV V DVOICHNOM PREDSTAVLENII~$U$.

{\sl (2)~bINOMIALXNOE RASPREDELENIE~$(t, p)$.\/} eSLI NEKOTOROE
SOBYTIE PROISHODIT S VEROYATNOSTXYU~$p$, I MY PROVODIM
$t$~NEZAVISIMYH ISPYTANIJ, POLNOE CHISLO~$N$ PROISHODYASHCHIH PRI |TOM SOBYTIJ
RAVNO~$n$ S VEROYATNOSTXYU~$\perm{t}{n}p^n(1-p)^{t-n}$ (SM.~P.~1.2.10). dLYA
|TOGO RASPREDELENIYA NET KAKOGO-LIBO PRYAMOGO METODA, ANALOGICHNOGO~(34).
oDNAKO MY MOGLI BY ISPOLXZOVATX TO OBSTOYATELXSTVO, CHTO ESLI $N_1$~IMEET
BINOMIALXNOE RASPREDELENIE~$(t_1, p)$ I ESLI, NEZAVISIMO, $N_2$~IMEET
BINOMIALXNOE RASPREDELENIE~$(t_2, p)$, TO $N_1+N_2$~IMEET BINOMIALXNOE
RASPREDELENIE~$(t_1+t_2, p)$. kOGDA $t$~VELIKO,
%%146
BINOMIALXNOE RASPREDELENIE PRIBLIZHENNO OPISYVAETSYA NORMALXNYM
RASPREDELENIEM SO SREDNIM~$tp$ I SREDNEKVADRATICHNYM
OTKLONENIEM~$\sqrt{tp(1-p)}$. sM. TAKZHE PRIEM, RASSMOTRENNYJ V UPR.~25.

{\sl (3)~rASPREDELENIE pUASSONA\/} SO SREDNIM ZNACHENIEM~$\mu$. eTO
RASPREDELENIE TAK ZHE SVYAZANO S |KSPONENCIALXNYM RASPREDELENIEM,
KAK BINOMIALXNOE S GEOMETRICHESKIM. oNO HARAKTERIZUET CHISLO REALIZACII V
EDINICU VREMENI SOBYTIJ, KAZHDOE IZ KOTORYH MOZHET PROIZOJTI V LYUBOJ MOMENT.
nAPRIMER, CHISLO IZLUCHAEMYH V SEKUNDU ALXFA-CHASTIC IMEET RASPREDELENIE
pUASSONA. vEROYATNOSTX TOGO, CHTO~$N=n$, RAVNA
$$
e^{-\mu}\mu^n/n!, \rem{$n\ge0$.}
\eqno(35)
$$

eSLI~$N_1$, $N_2$---NEZAVISIMYE PUASSONOVSKIE SLUCHAJNYE VELICHINY SO
SREDNIMI~$\mu_1$, $\mu_2$, TO VEROYATNOSTX TOGO, CHTO~$N_1+N_2=n$, RAVNA
$$
\sum_{0\le k \le n}{e^{-\mu_1}\mu_1^k\over k!} {e^{-\mu_2}\mu_2^{n-k}\over (n-k)!}
={e^{-(\mu_1+\mu_2)}(\mu_1+\mu_2)^n\over n!}.
$$
tAKIM OBRAZOM, $N_1+N_2$~IMEET RASPREDELENIE pUASSONA SO SREDNIM ZNACHENIEM~$(\mu_1+\mu_2)$.

pREDPOLOZHIM, CHTO MY HOTIM NAPISATX OBSHCHUYU PODPROGRAMMU, VYRABATYVAYUSHCHUYU
ZNACHENIYA PUASSONOVSKIH SLUCHAJNYH VELICHIN SO SREDNIM~$\mu$, GDE
$\mu$~ZADAETSYA PRI VHODE V PODPROGRAMMU.

\alg Q.(rASPREDELENIE pUASSONA S PROIZVOLXNYM~$\mu$.)

\st[vYCHISLITX |KSPONENTU.] pRISVOITX~$p\asg e^{-\mu}$ I~$N\asg0$, $q\asg1$.
(hOTYA $e^{-\mu}$~OBYCHNO VYCHISLYAETSYA S POMOSHCHXYU ARIFMETIKI S PLAVAYUSHCHEJ
TOCHKOJ S PRIVLECHENIEM STANDARTNOJ PODPROGRAMMY, VOZMOZHNO, RAZUMNEJ
POLXZOVATXSYA ARIFMETIKOJ S FIKSIROVANNOJ TOCHKOJ, PRAVILXNO VYBRAV
MASSHTAB I OKRUGLENIE DLYA POSLEDUYUSHCHIH OPERACIJ S~$p$ I~$q$.)

\st[pOLUCHITX SLUCHAJNOE CHISLO.] vYRABOTATX SLUCHAJNOE CHISLO~$U$, RAVNOMERNO
RASPREDELENNOE MEZHDU~$0$ I~$1$.

\st[uMNOZHITX.] uSTANOVITX~$q\asg qU$.

\st[pROVERITX, MENXSHE LI~$e^{-\mu}$.] eSLI~$q\ge p$, USTANOVITX~$N\asg N+1$
I VERNUTXSYA K SHAGU~\stp{2}. v PROTIVNOM SLUCHAE ALGORITM
ZAKANCHIVAETSYA VYVODOM~$N$.
\algend

chTOBY DOKAZATX SPRAVEDLIVOSTX METODA, ZAMETIM, CHTO NEZAVISIMYE
RAVNOMERNO RASPREDELENNYE SLUCHAJNYE VELICHINY UDOVLETVORYAYUT USLOVIYAM
$$
U_1\ge p, \quad U_1U_2\ge p, \quad, \ldots, \quad U_1U_2\ldots U_n\ge p,
\quad U_1U_2\ldots U_{n+1}<p
$$
%% 147
S VEROYATNOSTXYU
$$
p\left(\ln{1\over p}\right)^n\bigg/n! \rem{DLYA $0<p\le 1$.}
$$
pRIMENYAYA INDUKCIYU PO~$n$, POSREDSTVOM INTEGRIROVANIYA
$$
\int_p^1(p/u_1)(\ln (u_1/p))^{n-1}\,du_1/(n-1)!
$$
POLUCHIM NUZHNYJ REZULXTAT.

dRUGOJ METOD VYRABOTKI SLUCHAJNYH VELICHIN S RASPREDELENIEM pUASSONA,
PRINADLEZHASHCHIJ p.~kRIBS, OSNOVAN NA UPOMINAVSHEMSYA RANEE SVOJSTVE
SUMMIROVANIYA. pRI |TOM TREBUETSYA BOLEE TSHCHATELXNOE PROGRAMMIROVANIE, CHEM
PRI REALIZACII ALGORITMA~Q, NO PRI VYSOKOM UROVNE PROGRAMMIROVANIYA EGO
MOZHNO SDELATX BOLEE BYSTRYM.

\alg K.(rASPREDELENIE pUASSONA S PROIZVOLXNYM~$\mu$.) iSPOLXZUETSYA
VSPOMOGATELXNAYA TABLICA~$M[1]<M[2]<\cdots<M[n]$, OPISANNAYA NIZHE%
\note{1}{
v |TOM ALGORITME $n$~OZNACHAET SOVSEM NE TO, CHTO V PREDYDUSHCHIH ABZACAH I
ZADAETSYA DOSTATOCHNO PROIZVOLXNO; SM. NIZHE.---{\sl pRIM. PEREV.\/}}.

\st[nACHALXNAYA USTANOVKA.] uSTANOVITX~$m\asg \mu$, $j\asg n$, $N\asg 0$.

\st[$m\ge M[j]$?] eSLI~$m<M[j]$, PEREJTI K SHAGU~\stp{5}.

\st[vYRABOTATX DLYA SREDNEGO~$M[j]$.] vYRABOTATX ZNACHENIE~$X$ SLUCHAJNOJ
CELOCHISLENNOJ VELICHINY, IMEYUSHCHEJ RASPREDELENIE pUASSONA SO SREDNIM~$M[j]$.
(eTO MOZHNO |FFEKTIVNO SDELATX, ISPOLXZUYA ZARANEE SOSTAVLENNYE TABLICY V
SOOTVETSTVII S OBSHCHIM METODOM DLYA CELOCHISLENNYH RASPREDELENIJ,
RASSMOTRENNYM V UPR.~21.)

\st[iZMENITX~$N$, $m$.] uSTANOVITX~$m\asg m-M[j]$, $N\asg N+X$
I VERNUTXSYA K SHAGU~\stp{2}.

\st[uMENXSHITX~$j$.] uMENXSHITX~$j$ NA~$1$. eSLI~$j>0$, VERNUTXSYA K
SHAGU~\stp{2}, V PROTIVNOM SLUCHAE ALGORITM ZAKANCHIVAETSYA.
\algend

chTOBY ISPOLXZOVATX |TOT ALGORITM, MY DOLZHNY SOSTAVITX SPECIALXNYE
PROGRAMMY DLYA CHASTNYH ZNACHENIJ~$\mu$, ZADANNYH V TABLICE~$M[1]$, $M[2]$,~\dots,
$M[n]$.

nAPRIMER, MY MOGLI BY PRINYATX~$n=10$, TOGDA
$$
\vcenter{\halign{
\hfil$#$&${}#$\hfil\bskip&&\bskip$#$\hfil\bskip\cr
   j&=1       & 2       & 3      & 4      & 5      & 6      & 7 & 8 & 9 & 10\cr
M[j]&=2^{-15} & 2^{-12} & 2^{-9} & 2^{-6} & 2^{-3} & 2^{-1} & 1 & 2 & 4 & \hfill 8 \cr
}}
\eqno(36)
$$
%% 148

eTOT METOD NE|FFEKTIVEN DLYA BOLXSHIH ZNACHENIJ~$\mu$, SKAZHEM~$\mu\ge50$.
pRI~$\mu<M[1]=2^{-15}$ OPISANNYJ ALGORITM PRISVAIVAET~$N=0$, TAK KAK
VEROYATNOSTX TOGO, CHTO~$N>0$, RAVNA~$1-e^{-\mu}$, T.~E.\ MENXSHE~$1\over 32\,000$.
rASPREDELENIE pUASSONA DLYA MALYH ZNACHENIJ~$\mu$ MODELIROVATX
CHREZVYCHAJNO LEGKO, TAK KAK DLYA VSEH PRAKTICHESKIH CELEJ $N$~BUDET DOVOLXNO
MALO. tOLXKO DLYA RASPREDELENIJ S~$M[j]=4$ I~$8$ IZ PRIVEDENNOGO VYSHE
SPISKA ZNACHENIJ POTREBUYUTSYA BOLXSHIE TABLICY PRI VYPOLNENII SHAGA~K3.

dLYA BOLXSHIH ZNACHENIJ~$\mu$ aRENS PREDLOZHIL |FFEKTIVNYJ, NO DOVOLXNO
SLOZHNYJ METOD PORYADKA~$\sqrt{\mu}$. eGO PROCEDURA DELIT PUASSONOVSKOE
RASPREDELENIE NA DVE CHASTI, ODNA IZ KOTORYH NAPOMINAET RAVNOBEDRENNYJ TREUGOLXNIK.

\excercises
\ex[10] kAK VY PREDLOZHITE VYRABATYVATX SLUCHAJNYE CHISLA,
RAVNOMERNO RASPREDELENNYE MEZHDU SLUCHAJNYMI CHISLAMI~$\alpha$ I~$\beta$ ($\alpha<\beta$)?

\ex[M16] pREDPOLAGAYA, CHTO~$mU$---SLUCHAJNOE CELOE CHISLO MEZHDU~$0$ I~$m-1$,
NAJDITE \emph{TOCHNUYU} VEROYATNOSTX TOGO, CHTO~$\floor{kU}=r$, ESLI~$0\le r < k$.
sRAVNITE REZULXTAT S TREBUYUSHCHEJSYA VEROYATNOSTXYU~$1/k$.

\rex[14] oBSUDITE, CHTO POLUCHITSYA, ESLI TRAKTOVATX~$U$ KAK CELOE CHISLO I
VYRABATYVATX IZ NEGO SLUCHAJNOE CELOE MEZHDU~$0$ I~$k-1$, \emph{DELYA}~$U$
NA~$k$ VMESTO PREDLOZHENNOGO V TEKSTE UMNOZHENIYA. tAKIM OBRAZOM, (1)
SLEDUET IZMENITX TAK:
% PRIDETSYA STAVITX CR, TAK KAK TOKENY KONCA STROK UZHE ZASOSANY V ARGUMENTE
% S KATEGORIEJ NEAKTIVNYJ SIMVOL
$$
\vbox{
\mixcode
ENTA & 0 \cr
LDX  & U \cr
DIV  & K \cr
\endmixcode
}
$$
rEZULXTAT OKAZHETSYA V REGISTRE~$X$. hOROSHIJ LI |TO METOD?

\ex[m20] dOKAZHITE OBA SOOTNOSHENIYA V~(7).

\rex[21] pREDLOZHITE |FFEKTIVNYJ METOD VYCHISLENIYA SLUCHAJNOJ VELICHINY S
RASPREDELENIEM~$px+qx^2+rx^3$, GDE~$p\ge0$, $q\ge0$, $r\ge0$ I~$p+q+r=1$.

\rex[vm21] vELICHINA~$X$ VYCHISLYAETSYA SLEDUYUSHCHIM METODOM.
{\medskip\narrower
{\sl "shAG~1.\/}~vYRABOTATX DVA SLUCHAJNYH CHISLA~$U$, $V$.
   %
{\sl shAG~2.\/}~eSLI~$U^2+V^2\ge1$, VERNUTXSYA K SHAGU~1, INACHE
USTANOVITX~$X\asg U$."
\medskip}
\noindent kAKOVA FUNKCIYA RASPREDELENIYA~$X$? kAK CHASTO BUDET VYPOLNYATXSYA
SHAG~1? (oPREDELITE SREDNEE I SREDNEKVADRATICHNOE OTKLONENIE.)

\ex[M18] oB®YASNITE, POCHEMU V METODE mARSALXI DLYA NORMALXNYH SLUCHAJNYH
VELICHIN ZHELANIE VYBRATX~$p_j$ KRATNYMI~$1/256$ PRIVODIT K
FORMULE~$p_j=\floor{64 f(j/4)}/256$,  $1\le j \le 12$.

\ex[10] zACHEM OSOBO VYDELYATX UZKIE PRYAMOUGOLXNIKI~$f_{13}$,~\dots,
$f_{24}$ V METODE mARSALXI NARAVNE S BOLXSHIMI~$f_1$,~\dots, $f_{12}$?
(pOCHEMU |TO LUCHSHE, CHEM OB®EDINENIE KAZHDOJ PARY~$(f_1, f_{13})$,
$(f_2, f_{14})$,~\dots{} V ODIN BOLXSHOJ PRYAMOUGOLXNIK?)

\ex[vm10] pOCHEMU KRIVAYA~$f(x)$ NA RIS.~9 VYPUKLA VVERH PRI~$x<1$ I VNIZ
DLYA~$x>1$?

\ex[vm21] vYVEDITE FORMULY DLYA~$a_j$, $b_j$ V SOOTNOSHENII~(20). pOKAZHITE
TAKZHE, CHTO~$E[j]=16/j$, ESLI~$1\le j \le 4$; $E[j]=1/(e^{j/16-1/32}-1)$,
ESLI~$5\le j \le 12$.
%% 149

\rex[vm27] dOKAZHITE, CHTO SHAGI~M8--M9 V ALGORITME~M VYRABATYVAYUT ZNACHENIE
SLUCHAJNOJ VELICHINY, SOOTVETSTVUYUSHCHEJ HVOSTU NORMALXNOGO RASPREDELENIYA,
T.~E.\ PRI~$x\ge3$ VEROYATNOSTX TOGO, CHTO~$X<x$, RAVNA
$$
\int_3^x e^{-t^2/2}\,dt\bigg/\int_3^{\infty} e^{-t^2/2}\,dt.
$$

\ex[vm46] pOLINOM tEJCHROEVA~(22)---|TO USECHENNAYA SUMMA
CHEBYSHEVSKIH POLINOMOV, TAK CHTO NE MOZHET BYTX LUCHSHEGO PRIBLIZHENIYA POLINOMOM
STEPENI~$9$ K FUNKCII~$F^{-1}(F_1(R))$. mOZHNO LI ISPOLXZOVATX LUCHSHIJ POLINOM?

\ex[vm25] zADANO MNOZHESTVO $n$ NEZAVISIMYH NORMALXNYH SLUCHAJNYH VELICHIN
$X_1$, $X_2$,~\dots, $X_n$ SO SREDNIM~$0$ I DISPERSIEJ~$1$. pOKAZHITE, KAK
NAJTI KONSTANTY~$b_j$ I~$a_{ij}$, $1\le j \le i \le n$, TAKIE, CHTO ESLI
$$
Y_1=b_1+a_{11}X_1, \quad Y_2=b_2+a_{21}X_1+a_{22}X_2, \quad\ldots, \quad
Y_n=b_n+a_{n1}X_1+a_{n2}X_2+\cdots+a_{nn}X_n,
$$
TO~$Y_1$, $Y_2$,~\dots, $Y_n$---ZAVISIMYE, NORMALXNO RASPREDELENNYE
SLUCHAJNYE VELICHINY S ZADANNOJ MATRICEJ KOVARIACIJ~$(c_{ij})$, I KAZHDAYA
SLUCHAJNAYA VELICHINA~$Y_j$ IMEET SREDNEE~$\mu_j$. (kOVARIACIYA~$c_{ij}$
VELICHIN~$Y_i$ I~$Y_j$  OPREDELYAETSYA KAK SREDNEE
ZNACHENIE~$(Y_i-\mu_i)(Y_j-\mu_j)$. v CHASTNOSTI, $c_{jj}$~YAVLYAETSYA
DISPERSIEJ~$Y_j$, KVADRATOM SREDNEKVADRATICHNOGO OTKLONENIYA. nE VSYAKAYA
MATRICA~$(c_{ij})$ MOZHET BYTX MATRICEJ KOVARIACIJ, I PREDPOLAGAETSYA,
KONECHNO, CHTO VASH METOD DOLZHEN RABOTATX LISHX V TOM SLUCHAE, ESLI RESHENIE
ZADACHI SUSHCHESTVUET.)

\ex[M20] eSLI~$X$---SLUCHAJNAYA VELICHINA S NEPRERYVNOJ FUNKCIEJ
RASPREDELENIYA~$F(x)$, A~$c$---KONSTANTA, KAKOVO RASPREDELENIE~$cX$?

\ex[vm21] eSLI~$X_1$ I~$X_2$---NEZAVISIMYE SLUCHAJNYE VELICHINY S FUNKCIYAMI
RASPREDELENIYA~$F_1(x)$ I~$F_2(x)$ I PLOTNOSTYAMI~$f_1(x)=F_1'(x)$,
$f_2(x)=F_2'(x)$, TO KAKOVY FUNKCII RASPREDELENIYA I PLOTNOSTI VEROYATNOSTI
SLUCHAJNOJ VELICHINY~$X_1+X_2$?

\ex[vm25] (a)~pOKAZHITE, CHTO $\chi^2$-RASPREDELENIE S ODNOJ STEPENXYU
SVOBODY---|TO RASPREDELENIE DLYA~$X^2$, GDE~$X$---NORMALXNAYA SLUCHAJNAYA
VELICHINA SO SREDNIM, RAVNYM NULYU, I DISPERSIEJ, RAVNOJ EDINICE.
(b)~pOKAZHITE, CHTO $\chi^2$-RASPREDELENIE S DVUMYA STEPENYAMI SVOBODY---|TO
|KSPONENCIALXNOE RASPREDELENIE SO SREDNIM~$2$. (c)~pOKAZHITE, CHTO
ESLI~$X_1$ I~$X_2$---NEZAVISIMYE SLUCHAJNYE VELICHINY, IMEYUSHCHIE
$\chi^2$-RASPREDELENIE S~$\nu_1$ I $\nu_2$~STEPENYAMI SVOBODY, TO
$X_1+X_2$~IMEET $\chi^2$-RASPREDELENIE S $\nu_1+\nu_2$~STEPENYAMI SVOBODY.

\rex[m24] kAKOVA \emph{FUNKCIYA RASPREDELENIYA}~$F(x)$ DLYA GEOMETRICHESKOGO
RASPREDELENIYA S VEROYATNOSTXYU~$p$? kAKOVA \emph{PROIZVODYASHCHAYA
FUNKCIYA}~$G(z)$? chEMU RAVNY SREDNEE I SREDNEKVADRATICHNOE OTKLONENIE |TOGO
RASPREDELENIYA?

\ex[m24] pREDLOZHITE METOD VYCHISLENIYA SLUCHAJNOGO CELOGO CHISLA~$N$, TAKOGO,
CHTO ONO PRINIMAET ZNACHENIE~$n$ S VEROYATNOSTXYU~$np^2(1-p)^{n-1}$, $n\ge0$.
(oSOBYJ INTERES PREDSTAVLYAET SLUCHAJ, KOGDA $p$~OCHENX MALO.)

\ex[m22] (a)~nAJDITE, SKOLXKO RAZ V SREDNEM VYPOLNYAETSYA SHAG~E2 V
ALGORITME~E; CHEMU RAVNO SREDNEKVADRATICHNOE OTKLONENIE? (b)~tOT ZHE VOPROS DLYA SHAGA~E6.

\edef\exref{\the\excerno}
\rex[23] pREDPOLOZHIM, CHTO MY HOTIM VYCHISLITX SLUCHAJNOE ZNACHENIE
VELICHINY~$X$ S TAKIM DISKRETNYM RASPREDELENIEM:
$$
\vcenter{\halign{
\hfil$#$&\hfil${}#$\hfil\quad&&\hfil$#$\hfil\quad\cr
  \hbox{zNACHENIE }X=&x_1           & x_2          & x_3           & x_4          & x_5          & x_6 \cr
\hbox{vEROYATNOSTX} =& 90 \over 512 & 81 \over 512 & 131 \over 512 & 10 \over 512 & 32 \over 512 & 168 \over 512 \cr
}}
$$

pOKAZHITE, KAK MOZHNO |FFEKTIVNO VYCHISLYATX~$X$, ISPOLXZUYA DEVYATIRAZRYADNOE
DVOICHNOE SLUCHAJNOE CHISLO~$b_1b_2b_3b_4b_5b_6b_7b_8b_9$, OPREDELENNOE
ANALOGICHNO FORMULE~(16)
%% 150
S POMOSHCHXYU VSPOMOGATELXNYH TABLIC, SODERZHASHCHIH NE BOLEE 35~|LEMENTOV.

\rex[24] sITUACIYA V PREDYDUSHCHEM UPRAZHNENII NESKOLXKO IDEALIZIROVANA, POTOMU
CHTO OBYCHNO TAK NE BYVAET, CHTOBY VSE VEROYATNOSTI IMELI ODIN I TOT ZHE STOLX
UDOBNYJ MNOZHITELX, KAK~$1/512$. pREDPOLOZHIM, CHTO VEROYATNOSTI V
PREDYDUSHCHEM UPRAZHNENII BYLI IZMENENY, A V DEJSTVITELXNOSTI ONI DOLZHNY BYTX TAKIMI:
$$
\vcenter{\halign{
\hfil$#$&\hfil${}#$\hfil\quad&&\hfil$#$\hfil\quad\cr
  \hbox{zNACHENIE }X=& x_1                          & x_2                          & x_3                           & x_4                          & x_5                          & x_6 \cr
\hbox{vEROYATNOSTX} =& {89 \over 512}+\varepsilon_1 & {80 \over 512}+\varepsilon_2 & {131 \over 512}+\varepsilon_3 & {10 \over 512}+\varepsilon_4 & {32 \over 512}+\varepsilon_5 & {168 \over 512}+\varepsilon_6 \cr
}}
$$
zDESX~$0\le +\varepsilon_j <{1\over 512}$
I~$+\varepsilon_1++\varepsilon_2+\varepsilon_3+\varepsilon_4+\varepsilon_5+\varepsilon_6={2\over 512}$.
pOKAZHITE, KAK |FFEKTIVNO VYBRATX |TI CHISLA DLYA DANNOGO RAVNOMERNO
RASPREDELENNOGO SLUCHAJNOGO
%% ?? U=\cdot b_1 b_2
CHISLA~$U=b_1b_2\ldots b_t$, GDE $t$~SRAVNITELXNO VELIKO. kAK I V
UPR.~\exref, POPYTAJTESX ISPOLXZOVATX VSPOMOGATELXNYE TABLICY,
SODERZHASHCHIE NE BOLEE 35~|LEMENTOV.

\ex[vm46] mOZHNO LI POLUCHITX TOCHNOE PUASSONOVSKOE RASPREDELENIE
DLYA BOLXSHIH~$\mu$, VYRABOTAV SOOTVETSTVUYUSHCHUYU NORMALXNUYU SLUCHAJNUYU
VELICHINU I PREOBRAZOVAV EE KAKIM-NIBUDX UDOBNYM SPOSOBOM V CELOE CHISLO,
POLXZUYASX PRI |TOM (VOZMOZHNO USLOZHNENNOJ) POPRAVKOJ NA NEBOLXSHOJ PROCENT
VREMENI. mOZHNO LI PODOBNYM OBRAZOM REALIZOVATX BINOMIALXNOE RASPREDELENIE DLYA
BOLXSHIH~$t$?

\let\npar=\par
\ex[vm23] (dZH.~FON~nEJMAN.) eKVIVALENTNY LI SLEDUYUSHCHIE DVA
SPOSOBA VYRABOTKI SLUCHAJNOJ VELICHINY~$X$ (T.~E.\ BUDET LI VELICHINA~$X$
IMETX ODNO I TO ZHE RASPREDELENIE)?
{\medskip\narrower
{\sl mETOD~1.\/}~uSTANOVITX~$X\asg \sin((\pi/2)U)$, GDE~$U$---SLUCHAJNOE CHISLO.
 \npar
{\sl mETOD~2.\/}~vYRABOTATX DVA NEZAVISIMYH SLUCHAJNYH CHISLA~$U$, $V$
I, ESLI~$U^2+V^2\ge1$, POVTORITX TO ZHE SAMOE, POKA NE BUDET
VYPOLNYATXSYA~$U^2+V^2<1$. tOGDA POLOZHITX~$X\asg \abs{U^2-V^2}/(U^2+V^2)$.
\medskip}

\ex[vm40] (s.~uLAM, dZH.~FON~nEJMAN.) pUSTX~$V_0$---SLUCHAJNOE CHISLO
MEZHDU~$0$ I~$1$. oPREDELIM POSLEDOVATELXNOSTX~$\<V_n>$
SOOTNOSHENIEM~$V_{n+1}=4V_n\times(1-V_n)$. tEPERX, ESLI VYCHISLENIYA DELAYUTSYA
ABSOLYUTNO TOCHNO, REZULXTAT IMEET RASPREDELENIE~$\sin^2\pi U$,
GDE~$U$---RAVNOMERNO RASPREDELENNOE SLUCHAJNOE CHISLO. dRUGIMI SLOVAMI,
FUNKCIYA RASPREDELENIYA TAKOVA:
$$
F(x)={1\over \sqrt{2\pi}}\int_0^x {dx \over \sqrt{x(1-x)}}.
$$
v SAMOM DELE, ESLI MY NAPISHEM~$V_n=\sin^2 \pi U_n$, TO YASNO,
CHTO~$U_{n+1}=(2U_n)\bmod 1$. i IZ TOGO, CHTO POCHTI VSE DEJSTVITELXNYE CHISLA
IMEYUT SLUCHAJNOE DVOICHNOE PREDSTAVLENIE (SM.~\S~3.5), SLEDUET, CHTO
POSLEDOVATELXNOSTX~$U_n$ RAVNOMERNO RASPREDELENNAYA. nO ESLI
VYCHISLENIE~$V_n$ PROIZVODITSYA S KONECHNOJ TOCHNOSTXYU, |TI ARGUMENTY,
OKAZYVAYUTSYA NEVERNYMI, TAK KAK SKORO MY NACHINAEM IMETX DELO S SHUMOM OT
OSHIBOK OKRUGLENIYA (von Neumann, {\sl Collected Works,\/} Vol.~V, pp.~768--770).
pROVEDITE TEORETICHESKOE I |KSPERIMENTALXNOE (PRI RAZNYH
ZNACHENIYAH~$V_0$) ISSLEDOVANIE POSLEDOVATELXNOSTI~$\<V_n>$, OPREDELENNOJ
VYSHE, KOGDA VYCHISLENIYA PROVODYATSYA S KONECHNOJ TOCHNOSTXYU. pOHOZHE LI
RASPREDELENIE NA OZHIDAEMOE? mOZHNO LI KAK-NIBUDX ISPOLXZOVATX |TI CHISLA?

\ex[m25] pUSTX $X_1$, $X_2$~\dots, $X_5$---DVOICHNYE SLOVA, A KAZHDYJ IZ
DVOICHNYH RAZRYADOV NEZAVISIMO PRINIMAET ZNACHENIE~$0$ ILI~$1$ S
VEROYATNOSTXYU~$1/2$. kAKOVA VEROYATNOSTX TOGO, CHTO V DANNOJ POZICII
REZULXTAT~$X_1\lor (X_2\land (X_3\lor (X_4 \land X_5)))$ SODERZHIT~$1$.
sDELAJTE OBOBSHCHENIE.

%% 151
\bye