Gamma Dağılımında Parametrelere Göre Sayısal Değerlerin Bulunması

GAMMA DAGILIMINDA PARAMETRELERE

GÖRE SA ViSAL

DEGERLERİN

BULUNMASI

Prof.Dr. Erol YARIZ (*) Bahattin RÜZGAR ( **) GİRİŞ:

İstatistikte sürekli bir dağılım olan, Gamma _{dağılımına bulanın adına} atfen ERLANG kanunu da denir. ERLANG 1917 yılında telefon trafiğinde­ ki bekleyişleri ve trafiğin yoğunluğunu hat sayısının bir fonksiyonu olarak incelemek istediğinde, böyle bir dağılımı bulmuştur ((2) s. 66). Matematik Analizde Euler integrallerinin bir türü olan Gamma dağılımına aynı zaman-- -da Eulerien--kanunu adı da verilir.

1) GAMMA

FO~

JKSİYONUNUN İHTİMALLER TEORİSİNDEKİ YERİ,

GAMMA DAGILIMI:

. r(r)=

r

ı{

-1

e-x

dx

(1)

integraline ikinci türden Euler integrali veya Gamma fonksiyonu denir.

Ma-tematik istatistikte;

-TANIM:

X sürekli tesadüfi değişkeninin yalnız pozitif değerler aldığını varsaya-lım. X'in ihtimal yoğunluk fonksiyonu; · · - ·

f(x) =

.2:..

Qıx.f -

1 e-i..x (2) r(r)

şeklinde verilmişse, X tesadüfi değişkeninin dağılımına Gamma dağılımı denir.

GAMMA DAGILIMININ ÖZELLİKLERİ:

a) Bu dağılımın r

>O

ve

A.

>O olmak üzere iki parametresi vardır.

b) Gamma dağılımının r parametresinin bazı değerlerine göre grafiği aşağıdaki gibidir.

( x) Prof.Dr. M.Ü. İşletme Bölümü.

1.0

.75

.50

.25

A.

=

1

... _

·- -==-

--o

2

3

4 5

6

7

8

c) Gamma dağılımının matematik ümidi:

r

E(X)=

-A.

d) Gamma dağılımının varyansı:

r

V(X)=2 . A;

dir. ((5). s. 9 ve (1) s. 173).

e) Gamma dağılımının moment doğurucu fonksiyonu:

r

M{t)=(

2:..)

A.-t

x

dir. Moment doğurucu fonksiyondan matematiksel ümidi ve varyansı he-saplamak için, E(X)

=

M'(O) , (VX)

=

M" (0) - (M' (0))2 bağıntılarının yardımıyla M'(t)=

o!

'

.

o..-ır

1 E(X) =M' (0) =

f,

bulunur ((4) s. 113). M"(t) = r(r + 1

µ,r ,

<A-tr

2 V(X) = M"(O) - (M' (0))2=

..!..

.

.

A.2

f) Gamma dağılımının ihtimal yoğunluk fonksiyonunda (1 .2) r

=

1 için

f(x)

=

~'A:I.

bulunur. Bu ise Üstel dağılım, :.:\

=

1/2 ve r

=

n/2 için yazılırsa

il) PARAMETRELERE GÖRE SAYISAL DEGER TABLOLARININ

BULUNMASI:

Gamma dağilımının parametrelerine göre hesabı yapılırken, dağılımın

tanımından hareketle x >O,

A >

O ve r >O değerleri için ihtimal, beklenen

değer ve varyansı aşağıdaki programdan elde edilmiştir ve tablolar

bulun-muştur. Bu tablolardan bazıları aşağıda sunulmuş ve grafikleri yine bilgi-sayar ile çizilmiştir.

PROGRAM:

5 REM' GAMMA DAGILIMININ O < R ~ 1 O VE O <

A.

~ 1 O DEGERI iÇiN" 10 REM"IHTIMAL, MATEMATiKSEL ü~IT, VE VARYANSIN

BULUNMASI"

15 DIM a{10,10),B{10,10),C(10,10),D{10,10)

20 FK{0)=1: FOR R=1 TO

10:

FK(R)=FK(R-1)xR: NEXT 25 FOR X=1TO20

30 PRINr'

A.

R ihtimal matematik ümt varyans 35 PRINT'' _ _ _ _ 40 PRINT 45 FOR K=1TO10 50 D(K,0)=-1/EXP(XxK) 55 A(K,0)=-1/(KxEXP(XxK)) 60 D(K, 1 )=1 +D(K,O) 65 A(K, 1)=Kx(XxA(K,O)+1/KxA(K,O)+ 1/K"2) 70 FOR A=2TO10 75 D(K,R)=K" {R-1)/FK(R-1)xX"(8-1 )xD(K,O)+D(K,R-1) 80 A(K,R)=K" R/FK(R-1)xX"RxA(K,O)+R/(R-1)xA(K,R-1) 85 NEXT : NEXT 90 FOR K=1TO10 95 FOR R=1TO9

100 B(K,R)=R/KxA(K,R+ 1) : C(K,R)=B(K,R)-A(.K A)"2 105 NEXT R

110 NEXT K

115 FOR K=1 TO 10 120 FOR R=1TO10

125 PRINT "(";K;R")"; : PRINT" "; : PRINT USING "###.##########" ;

D(K,R); :PRINT" ";: PRINT USING "###.##########"; A(K,R);

:PRINT" "; : PRINT USING "###.##########" ; C(K,R):NEXT

:NEXT

130 PRINT 135

NEXT X

140 END

X=1

.

A.

r (1 1) (1 2) (1 3) (1 4) (1 5) (1 6) (1 7) (1 8) (1 9) (1 1 O) (2 1) (2 2) (2 3) (2 4) (2 5) (2 6) (2 7) (2 8) (2 9) (2 10) (3 1) (3 2) (3 3) (3 4) (3 5) (3 6) (3 7) (3 8) (3 9) (3 1 O) (4 1) (4 2) (4 3) (4 4) (4 5) ihtimal

o

.6321206000 0.2642411000 0.0803013800 0.0189881300 0.0036598220 0.0005941598 0.0000832161 0.0000102242 0.0000011002 0.0000000864 0.8646648000 0.5939941000 0.3233236000

o

.1428765000

o.

0526530200 0.0165636100 0.0045338020 0.0010967160 0.0002374441 0.0000464948 0.9502129000

o

.8008517000 0.5768099000 0.3527681000

o

.184 7367000

o.

9839178600 0.0335084400 0.0119044100 0.0038028950 0.0011023900

o.

9816843000

.

o.

9084218000 0.7618966000 0.5664298000' 0.3711629000 matematiksel ümit 0.2642411000 0.1606028000 0.0569644000 0.0246392900 0.0029708000 0.0004992977 0.0000715702 0.0000088025 0.0000007788 '

o

.0000001484 0.2969971000 0.3233236000 0.2143148000

o

.1 053060000 0.0414090200 0.0136014100 0.0038385090 0.0009497805 0.0002092314 0.0000415301 0.2669506000 0.3845399000 0.3527681000 0.2463156000

o

.1398632000 0.0670170100 0.0277771000 0.0101412200 0.0033073650

o

.00097 43462 0.2271055000

o

.3809483000 0.4248974000 0.3711630000 0.2685869000 varyans 0.0907793900 0.0881355600 0.0406729200 . 0.0116688500 0.0024876630 0.0004291717 0.0000616125 0.0000062306 0.0000013356 O. 0000000000 0.0734545300 0.1097767000

o.

1120282000 0.0717286700 0.0322888200 0.0113305300 0.0033094980 0.0008360235 0.0001868416 0.0000000000 0.0569173700 0.0873077800 0.1'218703000 0.1258129000 0.0921333000

o

.0510629200 0.0228912900 0.0087167960 0.0029121000

o

.0000000000 0.0436602000 0.0673270500

o

.0978344800

o

.1308250000 0.1353746000

x=1 matematiksel

A

r ihtimal ümit varyans

-(4 6) 0.2148694000 0.1660109000 0.1066659000 (4 7) 0.1106738000 0.0894836900 0.0667644000 (4 8) 0.0511333900 0.0427267000 0.0347691300 (4 9) 0.0213632100 0.0182973500 o. 0256384000 (4 10) 1 . 008132011 o 0.0070991970 0.0000000000 (5 1) 0.9932621000 0.1919145000 0.0331966800 (5 2) o. 9595723000 0.3501392000 0.0537963400 (5 3) 0.8753480000 0.4409845000 0.0740959800 (5 4) 0.7349741000 o .44 76054000 o .1068809000 (5 5) 0.5595067000 o .3840394000 0.1378937000 (5 6) 0.3840394000 o .2853799000 o .1426227000 (5 7) 0.2378165000 0.1867203000 o .1176653000 (5 8) 0.1333717000 0.1089498000 0.0797947600 (5 9) 0.0680936300 0.0572905200 0.0460208000 (5 10) o .0318280500 0.0273905600 0.0000000000 (6 1) 0.9975212000 0.1637748000 0.0252906600 (6 2) o. 9826488000 0.3126771000 0.0436990800 (6 3) o.9380312000 0.4243981000 0.0582007800 (6 4) 0.8487961000 0.4 766290000 0.0807805200 (6 5)

o.

7149435000 0.4619336000

o

.1146983000 (6 6) 0.5543203000 0.3936971000 0.1436927000 (6 7) 0.3036971000 0.2986901000 0.1484146000 (6 8) 0.2560201000 0.2036832000 o .1263609000 (6 9) 0.1527623000

o

.1258858000 0.0907046800 (6 1 O) 0.0839238000 0.0710346000 o .0000000000 (7 1) 0.9990881000 o. 1418150000 0.0194952000 (7 2) 0.9927049000 0.2772468000 0.0355711100 (7 3)

o.

9703638000 0.3935291000 0.0476675400 (7 4) 0.9182345000 0.4 725762000 0.0620965000 (7 5) 0.8270084000 0.4994941000 ' 0.0874172900 (7 6) 0.6992917000 0.4716763000 -0.1214811000 (7 7) 0.5502890000 0.4012862000 0.1485795000 (7 8) 0.4012862000 0.3096101000 0.1532090000 (7 9) 0.2709088000 0.2179339000 0.1334616000 (7 1 O) 0.1695041000 ·0.1407442000 0.0000000000 (8 1)

o.

9996646000 0.1246226000 0.0152894000 (8 2) 0.9969809000 0.2465615000

o

.0289842900. (8 3) 0.9862461000 0.3591074000 0.0398607700 (8 4) 0.9576199000 0.4501838000 0.0500732800 (8 5)

o.

9003676000

o

.5054 775000 0.0663483500 (8 6) 0.8087640000 0.5149692000

o.

0938010800 (8 7) 0.6866258000 0.4786592000

o

.127 4064000 (8 8) 0.5470391000 0.4074526000

o

.1527800000 (8 9) 0.4074526000 0.3187976000 . 0.1572785000

X=1

matematiksel

/\

r ihtimal ümit varyans

(8 10) 0.2833757000 0.2301426000

_o.

0000000000 (9 1) 0.9998766000 0.1109740000 0.0122222500 (9 2)

o.

9987659000 0.2208373000 0.0237326400 (9 3)

o.

9937679000 0.3262578000 0.0335612200 (9 4) 0.9787736000 0.4200162000

o

.0419344500 (9 5)

o.

9450364000

o

.4912830000 0.0524259100 (9 6)

o

.8843094000 0.5288128000 0.0709289300 (9 7)

o.

7932191000 0.5258578000 0.0998124500 (9 8) 0.6761030000

_o

.4838643000

o.

1326235000 (9 9) 0.54434 73000 0.4125916000

o

_{.15644 76000} (9 10) 0.4125916000 0.3266795000

o.

0000000000 . (10 1)

o.

9999546000 0.0999500600 0.0099545980 (10 2) 0.9995006000 0.1994461000 0.0196010900 (10 3) 0.9972306000

o

.2968992000 0.0283405600 (10 4) 0.9896639000 0.3882990000 0.0358067500 (10 5) 0.9707474000 0.4664571000 0.0433754300 (10 6) 0.9329141000 0.5219153000 0.0551118600 (10 7)

o

.8698586000 0.5458456000 0.0756735800 (10 8) 0.7797795000 . 0.5337443000 0.1054077000 (10 9) 0.6671805000 0.4878634000 0.1372537000 (10 1 O) 0.5420704000

_o

.4169604000

_o.

0000000000

X=2

İhtimal · matematiksel ümit varyans (1 1) 0.8646648000

_o.

5939941000 0.2938181000 (1 2) 0.5939941000 0.6466471000

o

.4391 066000 (1 3) 0.3233236000 0.4286296000 0.4481129000 (1 4) 0.1428765000 0.2106121000 0.2869147000 (1 5) 0.0526530200 0.0828180300 0.1291553000 (1 6) 0.0165636100

_o.

0272028200 0.0453221100 (1 7) 0.0045338020 0.0076770170 0.0132379900 (1 8) 0.0010967160 0.001899561

o

o.

0033440940 (1 9) 0.0002374441

o.

0004184628 0.0007473665 (1 1 O)

o.

0000444 7 40

o.

0000830602

_o.

0000000000 (2 1)

o.

9816843000

o

.4542109000

o.

17 46408000 (2 2)

o.

9084218000 0.7618967000 0.2693082000 (2 3) 0.7618966000 0.8497948000 0.3913379000 (2 4)

o.

5664298000

o.

7 423260000 0.5232998000 (2 5) 0.3711629000

_o

.5371739000 0.5414986000 (2 6) 0.2148694000 0.3320217000 0.4266637000 (2 7)

o.

11 06738000 0.1789674000 0.2670576000 (2 8) 0.0511333900

o.

0854534000 0.1390765000 (2 9) 0.0213632100 0.0365947000

o.

0625536000 (2 1 O) 0.008132011

o

0.0141983900

o.

0000000000 (3 1) 0.9975212000 0.3275496000 0.1011627000 (3 2)

o.

9826488000

o.

6253541 000 0.1747963000 (3 3) 0.9380312000 0.8487961000 0.2328031000

x=2

matematiksel

'A

r ihtimal ümit varyans

(3 4) 0.8487961000

o.

9532580000 0.3231221000 (3 5) O. 7149435000 0.9238671000 0.4587933000 (3 6) 0.5543203000

o.

7873943000 0.5747708000 (3 7) 0.3936971000 0.5973803000

o.

5936582000 (3 8) 0.2560201000 0.4073663000 0.5054436000 (3 9)

o.

1 527623000 0.2517716000 0.3628187000 (3 1 O)

o.

0839238000

o

.1420692000

o.

0000000000 (4 1)

o.

9996646000 0.2492452000 0.0611575800 (4

2)

0.9969809000

o

.4931230000 0.1159371000 (4 3) 0.9862461000 .

o.

7182149000 0.1594431000 (4 4) . 0.9576199000 0.9003676000 0.2002931000 (4 5) 0.9003676000 1 .

o

1 09550000 0.2653934000 (4 6) 0.8087640000 1 . 0299380000 0.3752043000 (4

7)

o.

6866258000 0.0573184000 0.5096255000 (4 8) 0.5470391000 0.8149051000 0.6111201000 (4 9) 0.4074526000 0.6375952000 0.6291141000 (4 1 O) 0.2833757000 0.4602853000

o

.0000000000 (5 1)

o.

9999546000 0.1999001000 0.0398183900 (5 2)

o.

9995006000 0.3988922000 0.0784043700 (5 3)

o.

9972306000 0.5937984000 0.1133623000 (5 4)

o.

9896639000

o.

7765980000 0.1432270000 (5 5) 0.9707474000 0.9329141000 0.1735017000 (5 6) 0.9329141000 1 . 043831 0000

o

.22044 7 4000 (5 7) 0.8698586000 1.0916910000 0.3026943000 (5 8) 0.7797795000 1.0674890000 0.4216306000 (5 9) 0.6671805000 0.9757268000 0.5490149000 (5 10) 0.5420704000 0.8339208000.

o.

0000000000 (6 1)

o.

9999938000

o.

1 666534000 0.0277532100 (6 2) 0.9999201000 0.3331593000 0.0552896100 (6 3)

o.

9994 777000 0.4988542000

o

.0819444400 (6 4)

o.

9977081000 0.6615998000 0.1065408000 (6 5)

o.

9923995000

o.

8163825000

o

.1286678000 (6 6) 0.9796588000 0.9541778000 0.1517896000 (6 7)

o.

9541776000 . 1.0622450000 0.1860372000 (6 8)

o.

91 04953000 1 . 1266300000 0.2459214000 (6 9) 0.8449720000 1 . 1364120000 0.3404949000 (6 1 O)

o.

7576076000 1.0879510000 0.0000000000 (7 1) 0.9999992000 0.1428554000 0.0204048400 (7 2) 0.9999875000 0.285687 4000 0;0407736000 (7 3)

o.

9999060000 0.4283681000 0.0609565700 (7 4) 0.9995258000 0.5703969000 0.0805526100 (7 5) 0.9981947000

o.

7103341000 0'.0989592700

x=2 _matematiksel

A.

r ihtimal ümit

_{__}

_{___}

_{v_~~Y.!~~}

_·-

_-

_--

---- ----

--- ---- ··-- --· -(7 6) 0.9944679000 0.8449473000 O. 1·151041000 (7 7) 0.9857720000

_o

_.

_{9683801 000 0.1341767000} (7 8), 0.9683802000 1 .0719370000 0.1595891000 (7 9) 0.9379448000 _{1 . 1 450580000 0.2028976000} (7 1 O) 0.8906006000 _{1 .1775980000}

_o.

_0000000000 (8 .1)

o.

9999999000 0.1249998000 0.0156245500 (8 2) 0.9999981000 0.2499959000 0.0312433000 (8 3) 0.9999837000 _{0.3749651000 0.0468261300} (8 4) 0.9999068000

_o

_.4997998000

_o.

_{06226 77700} (8 5) 0.9995995000 0.6241351000 _0.0773275200 (8 6)

o.

99861 62000

o

.

7 469954000 0.0916854100 (8 7) 0.9959939000 0.8662501000 0.1053719000 (8 8) 0.9900002000 0.9780126000 0.1197806000 (8 9) 0.9780127000 1.0762890000 0.1390133000 (8 10) 0.9567016000 _{1 . 1 532550000 0.0000000000} (9 1) 1.0000000000

o.

_{1111111 000 0.0123456200} (9 2) 0.9999997000 0.2222216000 0.0246903300 (9 3) 0.9999972000 _{0.3333275000 0.0370284800} (9 4)

o.

9999825000 0.4444070000

_o.

0493359600 (9 5)

o.

9999158000 0.5553755000 0.0615419500 (9 6)

o.

9996760000 0.6659710000 0.0735008100 (9 7)

o.

9989566000 O. 77552'71000

_o.

0850373500 (9 8) 0.9971066000 0~8-826168000 0.0962044000 (9 9)

o.

9929440000

_o.

9846188000 0.1078966000 (9 1 O)

o.

9846189000 1.0773710000

_o.

0000000000 (1

o

1) 1 .0000000000 0.0999999900

_o.

009999991

o

(10 2) 1.0000000000

_o

_.

1 999999000 0.0199998400 (10 3)

o

.

9999996000

_o.

2999990000

_o.

0299985600 (1

o

4)

o.

9999968000 0.3999932000

o .

_{.039991 0500} (10 5) b.9999831000 0.4999640000

_o.

0499594200 (10 6)

o.

9999281000 0.5998468000

_o.

0598566900 (1

o

7) 0.9997449000

_o.

_{6994549000 0.0695939100} (1

o

8) 0.9992214000

_o.

7983300000 0.0790723600 (10 9) 0.9979128000 0.8955039000 0.0883420100 (1

o

10) 0.9950046000 _0.9891881000

_o

_.

_0000000000

x=3 ihtimal · . matematiksel ümit varyans

(1 1) 0.9502129000 0.8008517000

o

.5122563000 (1 2) 0.8008517000 1 . 1536200000

_o.

7857696000 (1 3) 0.5768099000 1 .0583040000 1.0968320000 (1 4) 0.3527681000

_o

_.

7389466000 1.1323150000 (1 5) 0.1847367000 0.4195892000 0.8291975000 (1 6) 0.0839178600 0.2010505000 0.4595628000 (1 7)

o.

0335084400 0.0833306800 0.2060168000 (1 8) 0.0119044100 0.0304229600 0.0784448000 (1 9) 0.0038028950 0.0099212940

_o.

0262009000

X=3 matematiksel

A.

r ihtimal . ümit varyans

---· ····--· --- -·---·--··-·. -. ·-· ···---·---···--·---···---··-. ---·-·-- ----· ... --·--·--- ·-(1 10) 0.00·1102391

o

0.0029221480 0.0000000000 (2 1) 0.9975212000

o

.4913244000 0.2276160000 (2 2) 0.9826488000 O. 9380312000 0.3932917000 (2 3)

o

.

9380312000 1. 2731940000

o.

5238071000 (2 4) 0.8487961000 1 .42988 70000 O. 7270248000 (2 5)

o.

7149435000 1.3858010000 1.0322850000 . (2

₆₎

0.5543203000 1.1810910000 1.2932340000

(2

7) 0.3936971000 0.8960702000 1.3357310000 (2 8) 0.2560201000 0.6110493000 1.1372470000 (2 9) 0.1527623000 0.3776571000

o

.

81 63409000 (2 1 O) 0.0839238100 0.2131035000

o.

0000000000 (3 1) 0.9998766000 0.3329220000

o.

11 00003000 (3 2) 0.9987659000 0.6625119000 0.2135937000 (3 3) 0.9937679000 0.9787736000

o.

3020509000 (3 4) 0.9?87736000 1 .2600490000 0.3774098000 (3 5) 0.9450364000 1 .4 738490000 0.4718325000 (3 6)

o

.

8843094000 1 .5864380000 0.6383608000 (3 7) O. 7932191000 1.5775730000 0.8983114000 (3 8) 0.6761030000 1.4515930000 1.1936110000

(3

9) 0.5443473000 1 .237T750000 1 .4080290000 (3 10) 0.4125916000 0.9800382000

o.

0000000000 (4 1)

o.

9999938000 0.2499800000

o.

062444 7000 (4 2) 0.9999201000 0.4997389000 0.1244016000 (4 3) O. 9994 T77000 O. 7482811000 0.1843751000 (4 4) 0.9977081000 0.9923996000 0.2397167000 (4 5)

o

.

9923995000 1.2245740000 0.2895025000 (4 6) 0.9796588000 1 . 4 3·12660000 0.3415268000 (4 7) 0.9541776000 1.5933670000 0.4185844000 (4 8) 0.9104953000 1.6899440000

o.

5533233000 (4 9) 0.8449720000 1. 7046170000 0.7661135000 (4 1 O) O. 7576076000 1 . 6319260000

O.

0000000000 (5 1)

o.

9999996000

o

.

·1 999990000 0.0399972500 (5 2) 0.9999951000 0.3999843000 0.0799618700 (5 3) 0.9999607000 0.5998731000 0.1197410000 (5 4) 0.9997886000

o

.

7993146000 0.1588421000 (5 5)

o.

9991433000 0.9972074000 0.1964188000 (5

6)

0.9972075000 1 . 1908420000 0.2316524000 (5

l)

0.9923681000 1.3747970000 0.2660536000 (5 8) 0.9819978000 1 . 5400850000 0.3069580000 (5 9)

o.

9625534000 1.6742630000 0.3703706000 (5 10)

o.

9301462000 1.7630710000

o.

0000000000 (6 1) 1 . 0000000000

o

.1666666000 0.0277776400 (6 2) 0.9999997000

o.

3333324000

o.

0555532600

x=3 _matematiksel

A

r ihtimal ümit varyans

-(6 3) 0.9999972000 0.4999912000 0.0833141000 (6 4) 0.9999825000

_o.

6666106000 0.1110060000 (6 5) 0.9999158000

_o.

8330633000

o.

1384693000 (6 6) 0.9996760000

o.

9989566000 0.1653767000 (6 7)

o.

9989566000 1.1632910000 0.1913338000 (6 8) 0.9971066000 1 .3239250000 0.2164596000 (6 9) 0.9929440000 1 .4 769280000 0.2427671000 (6 10) 0.9846189000 1.6160560000

o.

0000000000 (7 1) 1 .0000000000 0.1428571000

o

.0204081600 (7 2) 1 .0000000000 0.2857142000 0.0408162000 (7 3) 0.9999998000 0.4285708000 0.0612231800 (7 4) 0.9999986000 0.5714243000 0.0816240000 (7 S) 0.9999925000 0.7142619000 . 0.1019992000 (7 6)

o.

9999667000 0.8570369000 0.1222924000 (7 7) 0.9998764000

o.

9996054000

o

.1423825000 (7 8) 0.9996054000 1.1415930000

_o

.1620889000 (7 9) 0.9988941000 1 .2821590000 0.1813229000 (710) 0.9972344000 1 . 4196420000

o.

0000000000 (8 1) 1 .0000000000

o

.1250000000 0.0156250000 (8 2) 1.0000000000

o

.2500000000 0.0312499900 (8 3) 1 .0000000000 0.3750000000 ' 0.0468749100 (8 4) 0.9999999000

o

.4999997000 0.0624993500 (8 5)

o.

9999994000 0.6249981000 0.0781212800 (8 6) 0.9999968000

o.

7 499903000

o.

0937335500 (8 7) 0.9999868000. 0.8749585000 0.1093160000 (8 8) 0.9999524000

o.

9998496000 0.1248224000 (8 9)

o.

9998494000 1 .1245220000 0.1401758000 (8 10) 0.99957 46000 1 .2486440000 0.0000000000 (9 1) 1.0000000000

o.

_{1111111 000 0.0123456800} (9 2) 1 .0000000000 0.2222222000 0.0246913600 (9 3) 1 .0000000000 0.3333334000 0.0370370300 (9 4) 1 .0000000000

o

.4444445000 0;0493826700 (9 5) 1 . 00000000.00 0.5555555000

o.

0617280900 (9 6) 0.9999998000 0.6666660000 0.0740724500 (9 7) 0.9999988000

o.

7777738000

o.

0864133200 (9 8)

o

.9999949000

o

.8888726000 0.0987429600 (9 9) 0.9999817000

o.

9999422000

o.

_{111 0441 000} (9 1 O)

o.

9999422000 1 .11 09280000

o.

0000000000 (10 1) 1 .0000000000 0.0999999900 0.0100000000 (10 2) 1 .0000000000 0.2000000000 0.0200000000 (10 3) 1 .0000000000 0.3000000000

o.

0300000100 (1

o

4) 1 .0000000000 0.4000000000

o.

0400000100 (10 5) 1 .0000000000 0.5000000000

o.

0499999800 (1

o

6) 1 . 0000000000 0.5999999000

o.

0599998600

x=3 ?\; r ihtimal (10 7)

o.

9999999000 (10 8)

o.

9999994000 (10 9) 0.9999979000 (10 1 O)

o.

9999929000 matematiksel . ümit 0.6999996000

o.

7999983000

o.

8999935000 0.9999776000 varyans

o.

0699994000 0.0799975400. 0.0899915700

o.

0000000000 ıı i 1 . 1 ıs

···---~---.,___

1

s

~

·~

.

.!

-..

j~

1 1

CD il >< ·-<

'I .._

..-:. i.:' il i~ 1' i ~~ .: , ... ,,

x ...: >< il c< ----~ ""' '·. ! ~ :~

s

1~

;;

['. ~ _ .. ---il ---_______ ..

---.

___________

~

- - - -

~

-~ 1 Q) .,.... .!.

~

~12

il

g

---

---~-~-~---~---~

-

-·

·

·

··

··

1 1 1 1 ,ı ... /

c~-===-~---3-~---~---SONUÇ:

Bilgisayar yardımıyla bu tür tabloların elde edilişinin basit bir programla

yapılabileceği görülmektedir.

A.

nın S'ten küçük değerleri için grafik ve

tablolarının daha anlamlı olduğu görülmektedir. Bu ise Poisson dağılımın­ da da

A,

nın O

>

A.

~ 5 aralığındaki anlamı ile çakışmaktadır. Sonuç

ola-rak Gamma dağılımının bir parametresi olan

X

nın Poisson dağılımının

parametresi olan

A.

ile aynı aralık için anlamlı olduğunu ortaya

koymak-tadır. Bazı parametrik değerlere göre dağılım fonksiyonunun asimptotik

özellik gösterdiği açıktır. ·

YARARLANILAN KAYNAKLAR

(1) ALLASIA, G. - lstatiscia Lavretto Bello, Torino, 1983.

(2) GÜNCE. E. - İstatistik Sözlüğü, ODTÜ - Ankara, 1970.

(3) MEYER, P. - lntroductory Probability and Statistical Applications,

Addi-son - Wesley Publ. New York, 1972.

(4) MOOD, A.M.; GRAYBILL, F.A.; BOES, D.C. - lntroduction to the The-ory of Statistics, lnternational Student Edition, Tokyo. 1974.

(5) YARIZ, E. - Gamma Fonksiyonunun Parametrelerinin Maksimum Ben-zerlik Yöntemi ile Tahmini, İstanbul, 1981.