Diszkrét egyenletes eloszlás

A valószínűségszámítás elméletében és a statisztika területén a diszkrét egyenletes eloszlás egy olyan valószínűségi eloszlás, ahol a felvehető értékek halmaza nem egy intervallum, hanem különálló számok halmaza, amelyek mind ugyanolyan valószínűséggel adódnak.

Egy közismert példa a diszkrét egyenletes eloszlásra a kockadobás, ahol a kocka teljesen szabályos. A lehetséges értékek 1, 2, 3, 4, 5, 6 és minden kockadobáskor bármely érték valószínűsége 1/6. Ha viszont két kockát dobunk, akkor az értékek összeadódnak, és az eloszlás már nem lesz egyenletes, mivel a két kocka dobása utáni ’eredmény’eknek már nem lesz egyenlő a valószínűsége. Ha egy [a,b] egész számokat tartalmazó tartományt tekintünk, akkor a és b lesznek az eloszlás fő paraméterei. Így a kumulatív eloszlásfüggvény k ∈ [a,b]-re:

A maximum becslése

k megfigyelésből, az egész számok között az N maximumot keressük (akarjuk megbecsülni). Ez a probléma úgy is ismert, mint a német tankprobléma, mely nyomán megbecsülték a német tankok számát a második világháború alatt. Az UMVU becslési módszer szerint:

ahol m a legnagyobb megfigyelt mintaelem, és k a minta nagysága.[1][2] A szórásnégyzet:[1]

így a szórás: közel ,

Jellemző görbék

Tömegfüggvény
Kumulatív eloszlásfüggvény

Irodalom

  • Johnson, Roger: Estimating the Size of a Population. (hely nélkül): Teaching Statistics 16. 1994.  

Kapcsolódó szócikkek

Források

  1. ↑ a b Johnson, Roger (1994), "Estimating the Size of a Population", Teaching Statistics 16 (2 (Summer)), DOI 10.1111/j.1467-9639.1994.tb00688.x
  2. ↑ Johnson, Roger (2006), "Estimating the Size of a Population", Getting the Best from Teaching Statistics, <http://www.rsscse.org.uk/ts/gtb/johnson.pdf>. Hozzáférés ideje: 2009-02-17 Archiválva 2008. április 10-i dátummal a Wayback Machine-ben Archivált másolat. [2008. november 20-i dátummal az eredetiből archiválva]. (Hozzáférés: 2013. február 7.)

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