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(fù)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(guān)，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(dāng)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BinomialDistribution提WT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(yīng)用學(xué)科大氣科學(xué)；氣候?qū)W；計算機科學(xué)目錄1定義?統(tǒng)計學(xué)定義?醫(yī)學(xué)定義2概念3性質(zhì)4圖形特點5應(yīng)用條件6應(yīng)用實例定義統(tǒng)計學(xué)定義在概率論和統(tǒng)計學(xué)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

2、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(dāng)稹二1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(chǔ)。醫(yī)學(xué)定義在醫(yī)學(xué)領(lǐng)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(dichotomousvariable),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(binomialdistribution)就是對這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(dāng)成功的概率(R)是恒定的，且各次試驗相互獨立，這種試驗在統(tǒng)計學(xué)

3、上稱為伯努利試驗(Bernoullitrial)。如果進行門次伯努利試驗，取得成功次數(shù)為=QN)的概率可用下面的二項分布概率公式來描述：P=C(X,n)*TtAX*(1-)A(n-X)=工bf上；小J75dM如“*建二項分布公式式中的n為獨立的伯努利試驗次數(shù)，兀為成功的概率，(1-兀)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)(binomialcoefficient)。所以的含義為：含量為n的樣本中，恰好有X例陽性數(shù)的概率。概念二項分布(BinomialDistribution),即重復(fù)n次的伯努利試驗(BernoulliExpe

4、riment)，用E表示隨機試驗的結(jié)果。，=上上(/>)*=b(kix,p)Qj1I-*Ifl),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(fù)試驗中發(fā)生K次的概率是P(E=K尸C(n,k)*pAk*(1-p)A(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(biāo)，表示的是方哥。那么就說這個屬于二項分布。其中P稱為成功概率。記作EB(n,p)期望：EE=np;方差：DE=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

5、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(shè)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：E(x;=EX(U+X+X+X(=«»方差：D(x=JDX(l)+X(2)+X(3).+X(«)=rtp(l-p)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(guān)；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

6、fit.ft二或削%g幅二項分布I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小時，而只允許k個式樣失敗，,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(fù)試驗n次，該事件發(fā)生k次的概率為：P=C(n,k)XpAkx(l-p)A(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(dāng)p=q時圖形是對稱的例如，上十"，p=q=1/2,各項的概率可寫作：p"+6p$?+15p】+20p、'

7、;+15p”/+fip1554-®=12 .當(dāng)pwq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pwq,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正態(tài)分布。故當(dāng)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(dāng)p<q且np>5,或p>q且nq>5,這時的n就用認為很大，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(biāo)準(zhǔn)差如果二項分布滿足p<q,np>5,(或p>q,np>5附，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

8、下性質(zhì)：=upx(5-10a)(7=的阿5-10b)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(biāo)準(zhǔn)差是根據(jù)理論推導(dǎo)而來的，故用科和b而不用X和S表示。它們的含意是指在二項試驗中，成功的次數(shù)的平均數(shù)科=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(科=np=),正面向上的散布程度為V10X(1/2)X(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

9、點(1)當(dāng)(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(dāng)(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的最大整數(shù)。應(yīng)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)的概率為兀，其對立結(jié)果的概率為1-兀，實際工作中要求兀是從大量觀察中獲得比較穩(wěn)定的數(shù)值。&=即+#歷S七項分布公式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

10、家族性等。應(yīng)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)分由猜測而造成的結(jié)果與真實的結(jié)果之間的界限，就要應(yīng)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p=q=i，2即猜對猜錯的概率各為0.5。曄=5故此二項分布接近正態(tài)分布：*二印二1。x0.5=5根據(jù)正態(tài)分布概率，當(dāng)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1.645(

11、7=5+L.645xL58=7.68它的意義是，完全憑猜測，10題中彳1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(yīng)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(8100.5)=10*9/2*0.58*0.52=45/1024b(9100.5)=10*0.59*0.51=10/1024b(10100.5)=1/1024根據(jù)概率加法，答8題及其以上的總概率為：45/1024+10/1024+1/1024

12、=56/1024=0.0547同理，可計算8題以下的I率為95%。(近似)nknkkP(1P)kPXx附表1二項分布表nxP0.0010.0020.0030.0050.010.020.030.050.100.150.200.250.30200.9980).99600.99400.99000.98010.960)40.9409)0.90250.81000.72250.(64000.55250.4900211.0000.00001.00001.C0000.99)990.999160.99910.99750.99000.97750.!96000.93750.9100300.9970).99400.9

13、9100.98510.97030.94120.912''0.85740.72900.61410J51200.42190.3430311.0000.00001.00000.99990.99)970.998180.997L0.99280.97200.93930.189600.84380.7840321.0000.0000100001.00000.99990.99900.9966i0.99200.98440.9730400.9960).99200.98810.98010.9(>060.922!40.8853，0.81450.65610.52200.，40960.31640.24

14、01411.0000.00000.99990.99990.99)940.99770.9948!0.98600.94770.89050.181920.73830.6517421.00001.00001.00001.00000.99990.99950.99630.98800.97280.94920.<)163431.00001.0000099990.99950.99)840.996510.991500.9950).99000.98510.97520.95,100.903i90.8587'0.77380.59050.44370.：32770.23730.1681511.0000.000

15、00.99990.99980.99)900.996>20.9915；0.97740.91850.83520.73730.63280.5282521.00001.00001.00000.99990.99970.99880.99140.97340.94210.89650J3369531.00001.0000100000.99950.99)780.99330.984,0.9692541.0000().9999099970.99900.9976600.9940).98810.98210.97040.941150.885i80.8330)0.73510.53140.37710.26210.1780

16、0.1176611.0000).99990.99990.99960.99850.99430.9875；0.96720.88570.77650.(65540.53390.4202621.0000.0000100001.(0000.99980.999150.9978!0.98420.95270.90110.83060.7443631.00001.0000099990.99870.99)410.983)00.962,0.9295641.00000.9999099960.99840.99)540.98911651.00001.0000099990.99980.9993700.9930).98610.9

17、7920.96550.93>210.868110.8080)0.69830.47830.32060.20970.13350.0824711.0000).99990.99980.99950.99)800.992!10.9829)0.95560.85030.71660J57670.44490.3294721.0000.0000100001.00000.99970.999110.9962!0.97430.92620.85200.75640.6471731.00001.0000099980.99730.98790.966>70.929,00.8740741.000010.999809988

18、0.99530.981710.9712751.0000(0.9999099960.99870.9962761.00001.0000099990.9998800.9920).98410.97630.96070.92270.850)80.7837'0.66340.43050.27250.16780.10010.0576811.0000).99990.99980.99930.99)730.989170.9777'0.94280.81310.65720J50330.36710.2553821.0000.0000100000.99990.99960.998170.9942!0.96190

19、.89480.79690.67850.5518831.00001.00000.99990.99960.99500.97860.9437'0.88620.8059k0841.00001.0000099960.99710.98960.97270.942()851.0000(0.9998099880.99580.9887861.00000.9999099960.9987871.00001.000009999900.9910().98210.97330.95590.91350.83370.7602:0.63020.38740.23160.13420.07510.0404911.00000).9

20、9990.99970.99910.99660.98(90.97180.92880.77480.59950.，43620.3)030.1960921.00001.00001.00000.99990.99940.99800.9916i0.94700.85910.73820.60070.4528931.0000.00000.99990.99940.99170.96610.91440.83430.7297941.00001.0000099910.99440.98040.95110.9012!950.9999(0.9994099690.99000.9747961.00001.0000099970.998

21、70.9957971.00000.999909996981.00001.00001000.99000).98020.97040.95110.90440.81710.73740.59870.34870.19690.10740.05630.02B21011.00000).99980.99960.99890.99570.98380.9655i0.91390.73610.54430.：37580.24400.14931021.00001.00001.00000.99990.99910.99720.9885i0.92980.82020.67780.52560.38281031.0000.00000.99

22、990.99900.98720.95000.879,0.77590.64961041.0000().9999099840.99010.96i720.92190.849'，1051.00000.9999099860.99360.98!030.952!71061.0000(0.9999099910.99650.98941071.00000.9999099960.99841081.00001.0000099991091.00001100.98910).97820.96750.94640.89530.800)70.7153i0.56880.31380.16730.08590.04220.019

23、81110.99990).99980.99950.99870.99480.98050.95870.89810.69740.49220.：32210.19710.11301121.00001.00001.00001.C0000.99980.99880.9963i0.98480.91040.77880.(61740.45520.31271131.0000.00000.99980.99840.98150.93060.838910.71330.56961141.0000().9999099720.98410.94960.885M0.789'，1151.00000.9997099730.9883

24、0.96>570.92,81161.0000(0.9997099800.99240.97841171.00000.9998099880.99571181.00000.9999099941191.00001.00001200.98810).97630.96460.94160.88640.78470.69380.54040.28240.14220.06870.03170.01381210.99990).99970.99940.99840.99380.97690.95140.88160.65900.44350.27490.15840.08501221.00001.00001.00001.000

25、00.99980.99850.9952:0.98040.88910.73580J55830.39070.25281231.0000().99990.99970.99780.97440.90780.7946i0.64880.49251241.00001.0000099980.9)9570.97610.92'40.842,00.72371251.00000.9995099540.98060.94560.882!21260.9999(0.9993099610.98570.96141271.00000.9999099940.99720.99051281.00000.9999099960.998

26、31291.00001.00000999812101.00001300.9871().97430.96170.93690.87750.76900.673010.51330.25420.12090.05500.02380.00971310.99990).99970.99930.99810.99280.97300.943(i0.86460.62130.39830.；23360.12670.06371321.00001.00001.00001.C0000.99970.99800.99380.97550.86610.69200J50170.33260.20251331.0000().99990.999

27、50.99690.96i580.88200.7473:0.58430.42061341.00001.0000099970.919350.96i580.90()90.794()0.65431351.00000.9991099250.970009980.83461360.99990.9987099300.97570.93761371.00000.9998099880.99440.98181381.00000.9998099900.99601391.00000.99990999313101.00000.999913111.00001400.98610).97240.95880.93220.86870

28、.75360.65280.48770.22880.10280.04400.01780.00681410.99990).99960.99920.99780.99160.96900.9355i0.84700.58460.35670.19790.1)100.04751421.00001.00001.00001.00000.99970.99750.9923i0.96990.84160.64790，44810.23110.16081431.0000().99990.99940.99580.95i590.85350.6982!0.52130.35521441.00001.0000099960.919080

29、.95i330.87()20.74150.58421451.00000.9985098850.95610.88830.780)51460.99980.9978098840.96170.90671471.00000.9997099760.98970.96851481.00000.9996099780.99171491.00000.99970998314101.00000.999814111.00001500.98510).97040.95590.92760.86010.73860.6333i0.46330.20590.08740.03520.01340.00471510.99990).99960

30、.99910.99750.99040.96470.927010.82900.54900.31860.16710.08020.03531521.00001.00001.00000.99990.99960.99700.9906i0.96380.81590.60420.：39800.23610.12681531.0000.0000099980.99920.99450.94440.8227'0.64820.46130.29691541.0000().9999099940.918730.93i830.83(i80.686；50.51551551.0000().9999099780.98320.9

31、3i890.85160.72161561.00000.9997099640.98190.94L340.868191571.00000.9994099580.98270.95001580.99990.9992099580.98481591.00000.9999099920.996315101.00000.99990999315111.00000.999915121.00001600.9841().96850.95310.92290.85150.72380.6143i0.44010.1853C.07430.02810.01000.00331610.99990).99950.99890.99710.

32、98910.96010.9182:0.81080.51470.28390.14070.03350.02611621.00001.00001.00000.99990.99950.99(30.98870.95710.78920.56140.：35180.19710.09941631.0000.00000.99980.99890.99300.93,60.789910.59810.40500.24591641.0000().9999099910.918300.921090.791!20.630；0.44991651.0000().9999099670.976509830.810)30.65981661

33、.00000.9995099440.97330.92!040.82471670.99990.9989099300.917290.92561681.00000.9998099850.99250.97431691.00000.9998099840.992916101.00000.99970998416111.00000.999716121.00001700.98310).96650.95020.91830.84290.70930.59580.41810.16680.06310.02250.00750.00231710.99990).99950.99880.99680.98770.95540.909

34、,0.79220.48180.25250.11820.05010.01931721.00001.00001.00000.99990.99940.99560.9866i0.94970.76180.51980.：30960.16370.07741731.0000.00000.99970.99860.99120.91740.7556i0.54890.35300.20191741.0000().9999099880.917790.90)130.751!20.573;00.3887,1751.0000().9999099530.96810.89)430.765530.59681761.00000.999

35、2099170.96230.89290.775i21770.99990.9983098910.95980.89541781.00000.9997099740.918760.95971791.00000.9995099690.987317101.00000.9999099940.996817111.00000.99990999317121.00000.999917131.00001800.98220).96460.94740.91370.83450.69510.578010.39720.15010.05360.01800.0)560.00161810.99980).99940.99870.996

36、40.98620.95050.89970.77350.45030.22410.09910.03950.01421821.00001.00001.00000.99990.99930.99480.9843i0.94190.73380.47970.；27130.13530.06001831.0000.0000099960.99820.98910.90,80.7202!0.50100.30570.16461841.0000().9998099850.917180.87940.71>40.518,00.33271851.0000().9998099360.95810.86i710.7150.534

37、41861.00000.9988098820.94870.86>100.72171870.99980.9973098370.94310.85931881.00000.9995099570.98070.94041890.99990.9991099460.979018101.00000.9998099880.993918111.00000.99980998618121.00000.999718131.00001900.9812().96270.94450.90920.82620.68120.5606)0.37740.13510.04560.01440.00420.00111910.9998(

38、).99930.99850.99600.98470.94540.8900)0.75470.42030.19850.08290.03100.01041921.0000.00001.00000.99990.99910.99390.981'0.93350.70540.44130.23690.11130.04621931.00001.00000.99950.99780.98680.88500.68410.45510.26310.13321941.00000.9998099800.96480.85i560.673，30.46540.28221951.00000.9998099140.94630.83份90.66780.473!91961.00000.9983098370.9)3240.8；>510.66(551970.99970.9959097670.9)22508801981.00000.9992099330.9)71309611990.99990.99840.99110.<)67419101.00000.99970.99770.<)89519111.00000.99