KURZUS: Matematika (Valószínűség-számítás és matematikai statisztika)

MODUL: Táblázatok

A standard normális eloszlás eloszlásfüggvényének (Fí) értékei

x Φ( x ) x Φ( x ) x Φ( x ) x Φ( x )
0,000,50000,500,69151,000,84131,500,9332
0,010,50400,510,69501,010,84381,510,9345
0,020,50800,520,69851,020,84611,520,9357
0,030,51200,530,70191,030,84851,530,9370
0,040,51600,540,70541,040,85081,540,9382
0,050,51990,550,70881,050,85311,550,9394
0,060,52390,560,71231,060,85541,560,9406
0,070,52790,570,71571,070,85771,570,9418
0,080,53190,580,71901,080,85991,580,9429
0,090,53590,590,72241,090,86211,590,9441
0,100,53980,600,72571,100,86431,600,9452
0,110,54380,610,72911,110,86651,610,9463
0,120,54780,620,73241,120,86861,620,9474
0,130,55170,630,73571,130,87081,630,9484
0,140,55570,640,73891,140,87291,640,9495
0,150,55960,650,74221,150,87491,650,9505
0,160,56360,660,74541,160,87701,660,9515
0,170,56750,670,74861,170,87901,670,9525
0,180,57140,680,75171,180,88101,680,9535
0,190,57530,690,75491,190,88301,690,9545
0,200,57930,700,75801,200,88491,700,9554
0,210,58320,710,76111,210,88691,710,9564
0,220,58710,720,76421,220,88881,720,9573
0,230,59100,730,76731,230,89071,730,9582
0,240,59480,740,77041,240,89251,740,9591
0,250,59870,750,77341,250,89441,750,9599
0,260,60260,760,77641,260,89621,760,9608
0,270,60640,770,77941,270,89801,770,9616
0,280,61030,780,78231,280,89971,780,9625
0,290,61410,790,78521,290,90151,790,9633
0,300,61790,800,78811,300,90321,800,9641
0,310,62170,810,79101,310,90491,810,9649
0,320,62550,820,79391,320,90661,820,9656
0,330,62930,830,79671,330,90821,830,9664
0,340,63310,840,79951,340,90991,840,9671
0,350,63680,850,80231,350,91151,850,9678
0,360,64060,860,80511,360,91311,860,9686
0,370,64430,870,80781,370,91471,870,9693
0,380,64800,880,81061,380,91621,880,9699
0,390,65170,890,81331,390,91771,890,9706
0,400,65540,900,81591,400,91921,900,9713
0,410,65910,910,81861,410,92071,910,9719
0,420,66280,920,82121,420,92221,920,9726
0,430,66640,930,82381,430,92361,930,9732
0,440,67000,940,82641,440,92511,940,9738
0,450,67360,950,82891,450,92651,950,9744
0,460,67720,960,83151,460,92791,960,9750
0,470,68080,970,83401,470,92921,970,9756
0,480,68440,980,83651,480,93061,980,9761
0,490,68790,990,83891,490,93191,990,9767
x Φ( x ) x Φ( x ) x Φ( x ) x Φ( x )
2,000,97722,500,99383,000,99873,500,9998
2,010,97782,510,99403,010,99873,510,9998
2,020,97832,520,99413,020,99873,520,9998
2,030,97882,530,99433,030,99883,530,9998
2,040,97932,540,99453,040,99883,540,9998
2,050,97982,550,99463,050,99893,550,9998
2,060,98032,560,99483,060,99893,560,9998
2,070,98082,570,99493,070,99893,570,9998
2,080,98122,580,99513,080,99903,580,9998
2,090,98172,590,99523,090,99903,590,9998
2,100,98212,600,99533,100,99903,600,9998
2,110,98262,610,99553,110,99913,610,9998
2,120,98302,620,99563,120,99913,620,9999
2,130,98342,630,99573,130,99913,630,9999
2,140,98382,640,99593,140,99923,640,9999
2,150,98422,650,99603,150,99923,650,9999
2,160,98462,660,99613,160,99923,660,9999
2,170,98502,670,99623,170,99923,670,9999
2,180,98542,680,99633,180,99933,680,9999
2,190,98572,690,99643,190,99933,690,9999
2,200,98612,700,99653,200,99933,700,9999
2,210,98642,710,99663,210,99933,710,9999
2,220,98682,720,99673,220,99943,720,9999
2,230,98712,730,99683,230,99943,730,9999
2,240,98752,740,99693,240,99943,740,9999
2,250,98782,750,99703,250,99943,750,9999
2,260,98812,760,99713,260,99943,760,9999
2,270,98842,770,99723,270,99953,770,9999
2,280,98872,780,99733,280,99953,780,9999
2,290,98902,790,99743,290,99953,790,9999
2,300,98932,800,99743,300,99953,800,9999
2,310,98962,810,99753,310,99953,810,9999
2,320,98982,820,99763,320,99953,820,9999
2,330,99012,830,99773,330,99963,830,9999
2,340,99042,840,99773,340,99963,840,9999
2,350,99062,850,99783,350,99963,850,9999
2,360,99092,860,99793,360,99963,860,9999
2,370,99112,870,99793,370,99963,870,9999
2,380,99132,880,99803,380,99963,880,9999
2,390,99162,890,99813,390,99973,890,9999
2,400,99182,900,99813,400,99973,901,0000
2,410,99202,910,99823,410,99973,911,0000
2,420,99222,920,99823,420,99973,921,0000
2,430,99252,930,99833,430,99973,931,0000
2,440,99272,940,99843,440,99973,941,0000
2,450,99292,950,99843,450,99973,951,0000
2,460,99312,960,99853,460,99973,961,0000
2,470,99322,970,99853,470,99973,971,0000
2,480,99342,980,99863,480,99973,981,0000
2,490,99362,990,99863,490,99983,991,0000
A táblázat használata

A táblázatban értelemszerűen az összetartozó x és Φ( x ) értékek szerepelnek egymás mellett. Mint látható, x oszlopában csak nem negatív értékek szerepelnek, míg Φ( x ) oszlopában csak 0,5 és annál nagyobbak. Mivel a standard normális eloszlás sűrűségfüggvénye szimmetrikus, ezért az így nem szereplő értékek esetén alkalmazhatjuk a Φ( x )=1Φ( x ) összefüggést.

Példák
Az alábbiakban ξ mindig standard normális eloszlású valószínűségi változót jelöl.

1. P( ξ<1 )=Φ( 1 )=0,8413
2. P( 0,3<ξ )=1P( ξ<0,3 )=1Φ( 0,3 )=10,6179=0,3821
3. P( ξ<a )=0,7 . Mivel Φ oszlopában nem szerepel 0,7, ezért a hozzá legközelebbi értéket választjuk, ami 0,6985. Az ehhez tartozó x érték pedig 0,52, így a0,52 .
4. P( ξ<0,2 )=Φ( 0,2 )=1Φ( 0,2 )=10,5793=0,4207
5. P( 1,2<ξ )=1P( ξ<1,2 )=1Φ( 1,2 )=1( 1Φ( 1,2 ) )=Φ( 1,2 )=0,8849
6. P( 2<ξ<1 )=Φ( 1 )Φ( 2 )=Φ( 1 )( 1Φ( 2 ) )=Φ( 1 )+Φ( 2 )1=
0,8413+0,97721=0,8185
7. P( 3<ξ<3 )=Φ( 3 )Φ( 3 )=Φ( 3 )( 1Φ( 3 ) )=2Φ( 3 )1=20,99871=0,9974