二項分布概念及圖表和查表方法

1、二項分布概念及圖表二項分布就是重復n次獨立的伯努利試驗。在每次試驗中只有兩種可能的結(jié)果，而且兩種結(jié)果發(fā)生與否互相對立，并且相互獨立，與其它各次試驗結(jié)果無關，事件發(fā)生與否的概率在每一次獨立試驗中都保持不變，則這一系列試驗總稱為n重伯努利實驗，當試驗次數(shù)為1時，二項分布服從0-1分布。中文名二項分布外文名Binomial Distribution提出者伯努利涉及實驗伯努利試驗；兩點分布屬    于概率論與數(shù)理統(tǒng)計應用學科大氣科學；氣候?qū)W；計算機科學目錄1 定義 統(tǒng)計學定義 醫(yī)學定義2 概念3 性質(zhì)4 

2、;圖形特點5 應用條件6 應用實例定義統(tǒng)計學定義在概率論和統(tǒng)計學中，二項分布是n個獨立的是/非試驗中成功的次數(shù)的離散概率分布，其中每次試驗的成功概率為p。這樣的單次成功/失敗試驗又稱為伯努利試驗。實際上，當  時，二項分布就是伯努利分布，二項分布是顯著性差異的二項試驗的基礎。醫(yī)學定義在醫(yī)學領域中，有一些隨機事件是只具有兩種互斥結(jié)果的離散型隨機事件，稱為二項分類變量（dichotomous variable），如對病人治療結(jié)果的有效與無效，某種化驗結(jié)果的陽性與陰性，接觸某傳染源的感染與未感染等。二項分布（binomial distribution）就是對這

3、類只具有兩種互斥結(jié)果的離散型隨機事件的規(guī)律性進行描述的一種概率分布?？紤]只有兩種可能結(jié)果的隨機試驗，當成功的概率（  ）是恒定的，且各次試驗相互獨立，這種試驗在統(tǒng)計學上稱為伯努利試驗（Bernoulli trial）。如果進行次伯努利試驗，取得成功次數(shù)為的概率可用下面的二項分布概率公式來描述：P=C(X,n)*X*(1-)(n-X)二項分布公式式中的n為獨立的伯努利試驗次數(shù)，為成功的概率，（1-）為失敗的概率，X為在n次伯努里試驗中出現(xiàn)成功的次數(shù)，表示在n次試驗中出現(xiàn)X的各種組合情況，在此稱為二項系數(shù)（binomial coefficient）。所以的含義為：含量為n的樣本

4、中，恰好有X例陽性數(shù)的概率。概念二項分布（Binomial Distribution），即重復n次的伯努利試驗（Bernoulli Experiment），用表示隨機試驗的結(jié)果。二項分布公式如果事件發(fā)生的概率是P,則不發(fā)生的概率q=1-p，N次獨立重復試驗中發(fā)生K次的概率是P(=K)= C(n,k) * pk * (1-p)(n-k)，其中C(n, k) =n!/(k!(n-k)!)，注意：第二個等號后面的括號里的是上標，表示的是方冪。那么就說這個屬于二項分布。其中P稱為成功概率。記作B(n,p)期望：E=np；方差：D=npq；其中q=1-p證明：由二項式分布的定義知，隨機變量X是n重伯努利

5、實驗中事件A發(fā)生的次數(shù)，且在每次試驗中A發(fā)生的概率為p。因此，可以將二項式分布分解成n個相互獨立且以p為參數(shù)的（0-1）分布隨機變量之和。設隨機變量X（k）(k=1,2,3.n)服從（0-1）分布，則X=X(1)+X(2)+X(3).X(n).因X(k)相互獨立，所以期望：方差：證畢。如果1在每次試驗中只有兩種可能的結(jié)果，而且是互相對立的；2每次實驗是獨立的，與其它各次試驗結(jié)果無關；3結(jié)果事件發(fā)生的概率在整個系列試驗中保持不變，則這一系列試驗稱為伯努利實驗。在這試驗中，事件發(fā)生的次數(shù)為一隨機事件，它服從二次分布。二項分布可二項分布以用于可靠性試驗?？煽啃栽囼灣３Ｊ峭度雗個相同的式樣進行試驗T小

6、時，而只允許k個式樣失敗，應用二項分布可以得到通過試驗的概率。若某事件概率為p，現(xiàn)重復試驗n次，該事件發(fā)生k次的概率為：P=C(n,k)×pk×(1-p)(n-k)。C(n,k)表示組合數(shù)，即從n個事物中拿出k個的方法數(shù)。性質(zhì)（一）二項分布是離散型分布，概率直方圖是躍階式的。因為x為不連續(xù)變量，用概率條圖表示更合適，用直方圖表示只是為了更形象些。1當p=q時圖形是對稱的例如，  ，p=q=1/2，各項的概率可寫作： 2當pq時，直方圖呈偏態(tài)，p<q與p>q的偏斜方向相反。如果n很大，即使pq，偏態(tài)逐漸降低，最終成正態(tài)分布，二項分布的極限分布為

7、正態(tài)分布。故當n很大時，二項分布的概率可用正態(tài)分布的概率作為近似值。何謂n很大呢?一般規(guī)定：當p<q且np5，或p>q且nq5，這時的n就被認為很大，可以用正態(tài)分布的概率作為近似值了。（二）二項分布的平均數(shù)與標準差如果二項分布滿足p<q，np5，(或p>q，np5)時，二項分布接近正態(tài)分布。這時，也僅僅在這時，二項分布的x變量(即成功的次數(shù))具有如下性質(zhì)：即x變量具有 = np，的正態(tài)分布。式中n為獨立試驗的次數(shù)，p為成功事件的概率，q=1- p。 由于n很大時二項分布逼近正態(tài)分布，其平均數(shù)，標準差是根據(jù)理論推導而來的，故用和而不用X和S表示。它們的含意是指在

8、二項試驗中，成功的次數(shù)的平均數(shù) = np ，成功次數(shù)的分散程 。例如一個擲10枚硬幣的試驗，出現(xiàn)正面向上的平均次數(shù)為5次(= np=)，正面向上的散布程度為10×（1/2）×（1/2)= 1.58(次)，這是根據(jù)理論的計算，而在實際試驗中，有的人可得10個正面向上，有人得9個、8個，人數(shù)越多，正面向上的平均數(shù)越接近5，分散程度越接近1.58 。圖形特點（1）當（n+1）p不為整數(shù)時，二項概率PX=k在k=(n+1)p時達到最大值；（2）當（n+1）p為整數(shù)時，二項概率PX=k在k=(n+1)p和k=(n+1)p-1時達到最大值。注：x為不超過x

9、的最大整數(shù)。 應用條件1各觀察單位只能具有相互對立的一種結(jié)果，如陽性或陰性，生存或死亡等，屬于兩分類資料。2已知發(fā)生某一結(jié)果（陽性）的概率為，其對立結(jié)果的概率為1-，實際工作中要求是從大量觀察中獲得比較穩(wěn)定的數(shù)值。二項分布公式3 n次試驗在相同條件下進行，且各個觀察單位的觀察結(jié)果相互獨立，即每個觀察單位的觀察結(jié)果不會影響到其他觀察單位的結(jié)果。如要求疾病無傳染性、無家族性等。應用實例二項分布在心理與教育研究中，主要用于解決含有機遇性質(zhì)的問題。所謂機遇問題，即指在實驗或調(diào)查中，實驗結(jié)果可能是由猜測而造成的。比如，選擇題目的回答，劃對劃錯，可能完全由猜測造成。凡此類問題，欲區(qū)分由猜測而造成

10、的結(jié)果與真實的結(jié)果之間的界限，就要應用二項分布來解決。下面給出一個例子。已知有正誤題10題，問答題者答對幾題才能認為他是真會，或者說答對幾題，才能認為不是出于猜測因素?分析：此題  ，即猜對猜錯的概率各為0.5。  ，故此二項分布接近正態(tài)分布：根據(jù)正態(tài)分布概率，當Z=1.645時，該點以下包含了全體的95%。如果用原分數(shù)表示，則為它的意義是，完全憑猜測，10題中猜對8題以下的可能性為95%，猜對8、9、10題的概率只5%。因此可以推論說，答對8題以上者不是憑猜測，而是會答。但應該明確：作此結(jié)論，也仍然有犯錯誤的可能，即那些完全靠猜測的人也有5%的可能性答

11、對8、9、10道題。此題的概率值，還可用二項分布函數(shù)直接計算，亦得與正態(tài)分布近似的結(jié)果：b(8 10 0.5)=10*9/2*0.58*0.52 = 45/1024b(9 10 0.5)=10*0.59*0.51 = 10/1024b(10 10 0.5) = 1/1024根據(jù)概率加法，答對8題及其以上的總概率為：45/1024+10/1024+1/1024=56/1024 = 0.0547 同理，可計算8題以下的概率為 95%。(近似) 附表 1二項分布表PX £ x = å n pk (1- p)n-kx k k =0 k nxp0.0010.0020.003

12、0.0050.010.020.030.050.100.150.200.250.30200.99800.99600.99400.99000.98010.96040.94090.90250.81000.72250.64000.56250.4900211.00001.00001.00001.00000.99990.99960.99910.99750.99000.97750.96000.93750.9100300.99700.99400.99100.98510.97030.94120.91270.85740.72900.61410.51200.42190.3430311.00001.00001.000

13、00.99990.99970.99880.99740.99280.97200.93930.89600.84380.7840321.00001.00001.00001.00000.99990.99900.99660.99200.98440.9730400.99600.99200.98810.98010.96060.92240.88530.81450.65610.52200.40960.31640.2401411.00001.00000.99990.99990.99940.99770.99480.98600.94770.89050.81920.73830.6517421.00001.00001.0

14、0001.00000.99990.99950.99630.98800.97280.94920.9163431.00001.00000.99990.99950.99840.99610.9919500.99500.99000.98510.97520.95100.90390.85870.77380.59050.44370.32770.23730.1681511.00001.00000.99990.99980.99900.99620.99150.97740.91850.83520.73730.63280.5282521.00001.00001.00000.99990.99970.99880.99140

15、.97340.94210.89650.8369531.00001.00001.00000.99950.99780.99330.98440.9692541.00000.99990.99970.99900.9976600.99400.98810.98210.97040.94150.88580.83300.73510.53140.37710.26210.17800.1176611.00000.99990.99990.99960.99850.99430.98750.96720.88570.77650.65540.53390.4202621.00001.00001.00001.00000.99980.9

16、9950.99780.98420.95270.90110.83060.7443631.00001.00000.99990.99870.99410.98300.96240.9295641.00000.99990.99960.99840.99540.9891651.00001.00000.99990.99980.9993700.99300.98610.97920.96550.93210.86810.80800.69830.47830.32060.20970.13350.0824711.00000.99990.99980.99950.99800.99210.98290.95560.85030.716

17、60.57670.44490.3294721.00001.00001.00001.00000.99970.99910.99620.97430.92620.85200.75640.6471731.00001.00000.99980.99730.98790.96670.92940.8740741.00000.99980.99880.99530.98710.9712751.00000.99990.99960.99870.9962761.00001.00000.99990.9998800.99200.98410.97630.96070.92270.85080.78370.66340.43050.272

18、50.16780.10010.0576811.00000.99990.99980.99930.99730.98970.97770.94280.81310.65720.50330.36710.2553821.00001.00001.00000.99990.99960.99870.99420.96190.89480.79690.67850.5518831.00001.00000.99990.99960.99500.97860.94370.88620.8059841.00001.00000.99960.99710.98960.97270.9420851.00000.99980.99880.99580

19、.9887861.00000.99990.99960.9987871.00001.00000.9999900.99100.98210.97330.95590.91350.83370.76020.63020.38740.23160.13420.07510.0404911.00000.99990.99970.99910.99660.98690.97180.92880.77480.59950.43620.30030.1960921.00001.00001.00000.99990.99940.99800.99160.94700.85910.73820.60070.4628931.00001.00000

20、.99990.99940.99170.96610.91440.83430.7297941.00001.00000.99910.99440.98040.95110.9012950.99990.99940.99690.99000.9747961.00001.00000.99970.99870.9957971.00000.99990.9996981.00001.00001000.99000.98020.97040.95110.90440.81710.73740.59870.34870.19690.10740.05630.02821011.00000.99980.99960.99890.99570.9

21、8380.96550.91390.73610.54430.37580.24400.14931021.00001.00001.00000.99990.99910.99720.98850.92980.82020.67780.52560.38281031.00001.00000.99990.99900.98720.95000.87910.77590.64961041.00000.99990.99840.99010.96720.92190.84971051.00000.99990.99860.99360.98030.95271061.00000.99990.99910.99650.98941071.0

22、0000.99990.99960.99841081.00001.00000.99991091.00001100.98910.97820.96750.94640.89530.80070.71530.56880.31380.16730.08590.04220.01981110.99990.99980.99950.99870.99480.98050.95870.89810.69740.49220.32210.19710.11301121.00001.00001.00001.00000.99980.99880.99630.98480.91040.77880.61740.45520.31271131.0

23、0001.00000.99980.99840.98150.93060.83890.71330.56961141.00000.99990.99720.98410.94960.88540.78971151.00000.99970.99730.98830.96570.92181161.00000.99970.99800.99240.97841171.00000.99980.99880.99571181.00000.99990.99941191.00001.00001200.98810.97630.96460.94160.88640.78470.69380.54040.28240.14220.0687

24、0.03170.01381210.99990.99970.99940.99840.99380.97690.95140.88160.65900.44350.27490.15840.08501221.00001.00001.00001.00000.99980.99850.99520.98040.88910.73580.55830.39070.25281231.00000.99990.99970.99780.97440.90780.79460.64880.49251241.00001.00000.99980.99570.97610.92740.84240.72371251.00000.99950.9

25、9540.98060.94560.88221260.99990.99930.99610.98570.96141271.00000.99990.99940.99720.99051281.00000.99990.99960.99831291.00001.00000.999812101.00001300.98710.97430.96170.93690.87750.76900.67300.51330.25420.12090.05500.02380.00971310.99990.99970.99930.99810.99280.97300.94360.86460.62130.39830.23360.126

26、70.06371321.00001.00001.00001.00000.99970.99800.99380.97550.86610.69200.50170.33260.20251331.00000.99990.99950.99690.96580.88200.74730.58430.42061341.00001.00000.99970.99350.96580.90090.79400.65431351.00000.99910.99250.97000.91980.83461360.99990.99870.99300.97570.93761371.00000.99980.99880.99440.981

27、81381.00000.99980.99900.99601391.00000.99990.999313101.00000.999913111.00001400.98610.97240.95880.93220.86870.75360.65280.48770.22880.10280.04400.01780.00681410.99990.99960.99920.99780.99160.96900.93550.84700.58460.35670.19790.10100.04751421.00001.00001.00001.00000.99970.99750.99230.96990.84160.6479

28、0.44810.28110.16081431.00000.99990.99940.99580.95590.85350.69820.52130.35521441.00001.00000.99960.99080.95330.87020.74150.58421451.00000.99850.98850.95610.88830.78051460.99980.99780.98840.96170.90671471.00000.99970.99760.98970.96851481.00000.99960.99780.99171491.00000.99970.998314101.00000.999814111

29、.00001500.98510.97040.95590.92760.86010.73860.63330.46330.20590.08740.03520.01340.00471510.99990.99960.99910.99750.99040.96470.92700.82900.54900.31860.16710.08020.03531521.00001.00001.00000.99990.99960.99700.99060.96380.81590.60420.39800.23610.12681531.00001.00000.99980.99920.99450.94440.82270.64820

30、.46130.29691541.00000.99990.99940.98730.93830.83580.68650.51551551.00000.99990.99780.98320.93890.85160.72161561.00000.99970.99640.98190.94340.86891571.00000.99940.99580.98270.95001580.99990.99920.99580.98481591.00000.99990.99920.996315101.00000.99990.999315111.00000.999915121.00001600.98410.96850.95

31、310.92290.85150.72380.61430.44010.18530.07430.02810.01000.00331610.99990.99950.99890.99710.98910.96010.91820.81080.51470.28390.14070.06350.02611621.00001.00001.00000.99990.99950.99630.98870.95710.78920.56140.35180.19710.09941631.00001.00000.99980.99890.99300.93160.78990.59810.40500.24591641.00000.99

32、990.99910.98300.92090.79820.63020.44991651.00000.99990.99670.97650.91830.81030.65981661.00000.99950.99440.97330.92040.82471670.99990.99890.99300.97290.92561681.00000.99980.99850.99250.97431691.00000.99980.99840.992916101.00000.99970.998416111.00000.999716121.00001700.98310.96650.95020.91830.84290.70

33、930.59580.41810.16680.06310.02250.00750.00231710.99990.99950.99880.99680.98770.95540.90910.79220.48180.25250.11820.05010.01931721.00001.00001.00000.99990.99940.99560.98660.94970.76180.51980.30960.16370.07741731.00001.00000.99970.99860.99120.91740.75560.54890.35300.20191741.00000.99990.99880.97790.90

34、130.75820.57390.38871751.00000.99990.99530.96810.89430.76530.59681761.00000.99920.99170.96230.89290.77521770.99990.99830.98910.95980.89541781.00000.99970.99740.98760.95971791.00000.99950.99690.987317101.00000.99990.99940.996817111.00000.99990.999317121.00000.999917131.00001800.98220.96460.94740.9137

35、0.83450.69510.57800.39720.15010.05360.01800.00560.00161810.99980.99940.99870.99640.98620.95050.89970.77350.45030.22410.09910.03950.01421821.00001.00001.00000.99990.99930.99480.98430.94190.73380.47970.27130.13530.06001831.00001.00000.99960.99820.98910.90180.72020.50100.30570.16461841.00000.99980.9985

36、0.97180.87940.71640.51870.33271851.00000.99980.99360.95810.86710.71750.53441861.00000.99880.98820.94870.86100.72171870.99980.99730.98370.94310.85931881.00000.99950.99570.98070.94041890.99990.99910.99460.979018101.00000.99980.99880.993918111.00000.99980.998618121.00000.999718131.00001900.98120.96270.

37、94450.90920.82620.68120.56060.37740.13510.04560.01440.00420.00111910.99980.99930.99850.99600.98470.94540.89000.75470.42030.19850.08290.03100.01041921.00001.00001.00000.99990.99910.99390.98170.93350.70540.44130.23690.11130.04621931.00001.00000.99950.99780.98680.88500.68410.45510.26310.13321941.00000.99980.99800.96480.85560.67330.46540.28221951.00000.99980.99140.94630.83690.66780.47391961.00000.99830.98370.93240.82510.66551970.99970.99590.97670.92250.81801981.00000.99920.99330.97130.91611990.99990.99840.99110.967419101.00000.99970.99770.989519111.00000.99