Binomial vs Normal Distribution
Qhov muaj feem cuam tshuam ntawm qhov sib txawv ntawm qhov sib txawv ua lub luag haujlwm tseem ceeb hauv kev txheeb cais. Tawm ntawm qhov kev faib tawm qhov tshwm sim, qhov kev faib tawm binomial thiab kev faib ib txwm yog ob qhov tshwm sim feem ntau hauv lub neej tiag tiag.
Dab tsi yog kev faib tawm binomial?
Binomial faib yog qhov muaj peev xwm faib tau raws li qhov sib txawv ntawm qhov sib txawv X, uas yog tus naj npawb ntawm kev ua tiav ntawm qhov kawg ntawm kev ywj pheej muaj / tsis muaj kev sim txhua qhov uas muaj qhov tshwm sim ntawm kev vam meej p. Los ntawm lub ntsiab txhais ntawm X, nws yog pov thawj tias nws yog ib tug discrete random variable; Yog li ntawd, binomial faib yog discrete ib yam nkaus.
Kev faib tawm yog suav tias yog X ~ B (n, p) qhov twg n yog tus lej ntawm kev sim thiab p yog qhov ua tau zoo. Raws li qhov kev xav tau, peb tuaj yeem txiav txim siab tias B (n, p) ua raws li qhov tshwm sim ntawm qhov ua haujlwm loj [latex] B(n, p)\\sim \\binom{n}{k} p^{k} (1-p)^{(n-k)}, k=0, 1, 2, …n [/latex]. Los ntawm qhov sib npaug no, nws tuaj yeem txiav txim siab ntxiv tias tus nqi xav tau ntawm X, E(X)=np thiab qhov sib txawv ntawm X, V(X)=np (1- p).
Piv txwv li, xav txog kev sim random ntawm pov ib npib 3 zaug. Txhais kev vam meej raws li tau txais H, tsis ua tiav raws li tau txais T thiab qhov sib txawv tsis sib xws X raws li tus naj npawb ntawm kev ua tiav hauv qhov kev sim. Tom qab ntawd X ~ B (3, 0.5) thiab qhov tshwm sim loj ua haujlwm ntawm X muab los ntawm [latex] \binom{3}{k} 0.5^{k} (0.5)^{(3-k)}, k=0, 1, 2[/latex]. Yog li, qhov tshwm sim ntawm qhov tsawg kawg yog 2 H's yog P (X ≥ 2)=P (X=2 lossis X=3)=P (X=2) + P (X=3)= 3 C2(0.52)(0.51) + 3 C3(0.53)(0.50)=0.375 + 0.125=0.5.
Dab tsi yog kev faib khoom ib txwm?
Ib txwm faib yog qhov kev faib tawm tas li uas tau txiav txim siab los ntawm qhov muaj peev xwm ceev ua haujlwm, [latex] N(\mu, \\sigma)\\sim\\frac{1}{\sqrt{2 \\pi \\sigma^{2}}} / e^{- \\frac{(x-\\mu)^{2}}{2 \\sigma^{2}}} [/latex]. Cov tsis muaj [latex] \mu thiab \\sigma [/latex] qhia qhov txhais tau tias thiab tus qauv sib txawv ntawm cov pej xeem nyiam. Thaum [latex] \mu=0 thiab \\sigma=1 [/latex] qhov kev faib tawm yog hu ua tus qauv kev faib tawm.
Qhov kev faib tawm no yog hu ua ib txwm vim feem ntau ntawm cov xwm txheej ntuj ua raws li kev faib tawm ib txwm. Piv txwv li, IQ ntawm tib neeg cov pejxeem feem ntau faib. Raws li pom los ntawm daim duab nws yog unimodal, symmetrical ntawm lub ntsiab lus thiab tswb zoo li tus. Qhov nruab nrab, hom, thiab nruab nrab yog coinciding. Thaj chaw hauv qab txoj kab nkhaus sib haum rau feem ntawm cov pej xeem, txaus siab rau ib qho xwm txheej.
Qhov feem ntawm cov pejxeem hauv lub sijhawm [latex] (\mu – \\sigma, \\mu + \\sigma) [/latex], [latex] (\mu – 2 \\sigma, \\mu + 2 \\sigma) [/latex], [latex] (\mu – 3 \\sigma, \\mu + 3 \\sigma) [/latex] yog kwv yees li 68.2%, 95.6% thiab 99.8% raws.
Dab tsi yog qhov txawv ntawm Binomial thiab Kev faib tawm ib txwm?
- Binomial faib yog ib qho kev faib tawm qhov tshwm sim thaum qhov kev faib tawm ib txwm yog ib qho txuas ntxiv.
- Qhov tshwm sim muaj nuj nqi ntawm qhov kev faib tawm binomial yog [latex]B(n, p)\\sim \\binom{n}{k} p^{k} (1-p)^{(n-k) } [/latex], whereas qhov tshwm sim qhov ceev ntawm qhov kev faib tawm ib txwm yog [latex] N(\mu, \\sigma)\\sim\\frac{1}{\sqrt{2 \\pi \\sigma ^{2}}} / e^{- \\frac{(x-\\mu)^{2}}{2 \\sigma^{2}}} [/latex]
- Binomial faib yog kwv yees nrog ib txwm faib raws li qee yam mob tab sis tsis yog lwm txoj hauv kev.
