Cov Lus Nug Piv Txwv Sib Tham Txog Kev Ua Haujlwm ntawm Binomial Distribution
Qhov kev faib tawm binomial yog qhov kev faib tawm qhov muaj feem cuam tshuam uas piav qhia txog tus lej ntawm kev ua tiav hauv kev sim uas muaj ntau qhov kev sim ywj pheej nrog ob qhov tshwm sim: kev ua tiav thiab kev ua tsis tiav. Txhua qhov kev sim hu ua kev sim, thiab qhov kev faib tawm binomial feem ntau siv rau hauv cov xwm txheej uas tus lej ntawm kev ua tiav thoob plaws ntau qhov kev sim ywj pheej yog qhov txaus siab. Hauv tsab xov xwm no, peb yuav tham txog cov ntsiab lus tseem ceeb ntawm kev faib tawm binomial thiab muab cov piv txwv thiab cov kev daws teeb meem.
Cov Ntsiab Lus Tseem Ceeb ntawm Binomial Distribution Function
Ua ntej peb nkag mus rau cov lus nug piv txwv thiab kev sib tham, cia peb tham txog qee lub ntsiab lus tseem ceeb ntsig txog kev faib tawm binomial.
1. Kev Txhais: Qhov kev faib tawm binomial yog txhais tias yog qhov sib sau ua ke ntawm kev ua tiav hauv 'n' kev sim ywj pheej, qhov twg txhua qhov kev sim muaj ob qho txiaj ntsig: kev ua tiav (nrog rau qhov muaj feem p) lossis kev ua tsis tiav (nrog rau qhov muaj feem q = 1 - p).
2. Kev Ua Haujlwm Feem Puas: Lub luag haujlwm feem puas ntawm kev faib tawm binomial yog:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{nk}
\]
Qhov twg:
- \( P(X = k) \) yog qhov muaj feem yuav muaj k qhov kev vam meej hauv n qhov kev sim.
- \( \binom{n}{k} \) yog kev sib xyaw ua ke ntawm n coj k, uas yog txhais tias yog \( \frac{n!}{k!(nk)!} \).
- \( p \) yog qhov feem pua ntawm kev vam meej ntawm txhua qhov kev sim siab.
- \( (1-p) \) yog qhov feem pua ntawm kev ua tsis tiav ntawm txhua qhov kev sim.
3. Tus Nqi Xav Tau thiab Kev Hloov Pauv:
– Tus nqi xav tau (qhov nruab nrab) ntawm kev faib tawm binomial yog \( \mu = np \).
- Qhov variance ntawm binomial faib yog \( \sigma^2 = np(1-p) \).
Tam sim no, cia peb siv cov tswv yim no rau hauv ib qho piv txwv teeb meem kom muab kev nkag siab tob dua.
Piv txwv lus nug 1: Kev suav lej yooj yim ntawm Binomial Distribution
Lo lus nug:
Ib lub tuam txhab tsim cov khoom siv hluav taws xob nrog qhov muaj feem yuav 0.95 uas txhua yam khoom yuav dhau qhov kev xeem zoo. Yog tias muaj 10 yam khoom raug tsim tawm, xam qhov muaj feem yuav dhau 8 yam khoom.
Kev Sib Tham:
Peb siv tau cov mis faib tawm binomial los daws qhov teeb meem no. Ua ntej, peb txheeb xyuas cov kev teeb tsa hauv qab no:
– \( n \) (tag nrho cov kev sim) = 10
– \( k \) (tus naj npawb ntawm kev ua tiav) = 8
– \( p \) (qhov feem pua ntawm kev vam meej) = 0.95
– \( q \) (qhov yuav ua tsis tiav) = 1 – 0.95 = 0.05
Tom qab ntawd hloov cov nqi no rau hauv cov mis faib binomial:
\[
P(X = 8) = \binom{10}{8} (0.95)^8 (0.05)^2
\]
Ua ntej, xam qhov sib xyaw ua ke \( \binom{10}{8} \):
\[
\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 \times 9 \times 8!}{8! \times 2!} = \frac{10 \times 9}{2 \times 1} = 45
\]
Tom qab ntawd, xam cov feem pua \( (0.95)^8 \) thiab \( (0.05)^2 \):
\[
(0.95)^8 \kwv yees li 0.6634
\]
\[
(0.05)^2 = 0.0025
\]
Thaum kawg, muab tag nrho cov nqi ntawd sib npaug kom tau txais:
\[
P(X = 8) = 45 x 0.6634 x 0.0025 x kwv yees li 0.0744
\]
Yog li, qhov muaj feem uas 8 ntawm 10 yam khoom yuav dhau qhov kev xeem zoo yog li ntawm 0.0744 lossis 7.44%.
Piv txwv lus nug 2: Qhov muaj feem yuav sib sau ua ke
Lo lus nug:
Tseem nrog tib lub tuam txhab, xam qhov muaj feem uas tsawg kawg yog 9 ntawm 10 yam khoom yuav dhau qhov kev xeem zoo.
Kev Sib Tham:
Yuav kom daws tau qhov teeb meem no, peb yuav tsum xam qhov feem pua ntawm kev sib sau ua ke. Qhov feem pua uas tsawg kawg yog 9 ntawm 10 yam khoom dhau qhov kev xeem txhais tau tias peb xam \( P(X \geq 9) \), uas tuaj yeem sau ua:
\[
P(X\geq 9) = P(X = 9) + P(X = 10)
\]
Siv cov mis faib tawm binomial:
\[
P(X = 9) = \binom{10}{9} (0.95)^9 (0.05)^1
\]
\[
P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^0
\]
Ua ntej, xam qhov sib xyaw ua ke rau txhua rooj plaub:
\[
\binom{10}{9} = \frac{10}{9!(10-9)!} = 10
\]
\[
\binom{10}{10} = 1
\]
Tom qab ntawd, xam cov feem pua rau \( P(X = 9) \) thiab \( P(X = 10) \):
\[
P(X = 9) = 10 \times (0.95)^9 \times 0.05
\]
\[
(0.95)^9 \kwv yees li 0.6302
\]
\[
P(X = 9) = 10 x 0.6302 x 0.05 x kwv yees li 0.3151
\]
\[
P(X = 10) = 1 \times (0.95)^{10} \times 1
\]
\[
(0.95)^{10} \kwv yees li 0.5987
\]
\[
P(X = 10) = 0.5987
\]
Tag nrho qhov muaj feem rau \( P(X \geq 9) \):
\[
P(X \geq 9) = 0.3151 + 0.5987 \approx 0.9138
\]
Yog li, qhov muaj feem yuav tsawg kawg yog 9 ntawm 10 yam khoom dhau qhov kev xeem zoo yog li ntawm 0.9138 lossis 91.38%.
Piv txwv lus nug 3: Tus nqi xav tau thiab kev hloov pauv
Lo lus nug:
Xam tus nqi xav tau thiab qhov sib txawv ntawm tus lej ntawm cov khoom uas dhau qhov kev xeem zoo ntawm 10 yam khoom uas tsim tawm, nrog rau qhov muaj feem yuav dhau ntawm 0.95.
Kev Sib Tham:
Siv cov mis no:
– Tus nqi xav tau (nruab nrab) \( \mu = np \)
– Kev Hloov Pauv \( \sigma^2 = np(1-p) \)
Nrog \(n = 10 \) thiab \(p = 0.95 \):
\[
\mu = 10 \times 0.95 = 9.5
\]
\[
\sigma^2 = 10 x 0.95 x 0.05 = 0.475
\]
Yog li, tus nqi xav tau ntawm tus lej ntawm cov khoom uas dhau qhov kev xeem zoo yog 9.5, thiab qhov sib txawv yog 0.475.
Xaus
Los ntawm peb qhov piv txwv teeb meem saum toj no, peb tau tham txog yuav ua li cas xam qhov muaj feem cuam tshuam siv kev faib tawm binomial rau ntau qhov xwm txheej: xam qhov muaj feem cuam tshuam tseeb, qhov muaj feem cuam tshuam sib sau ua ke, thiab xam tus nqi xav tau thiab qhov sib txawv. Kev paub txog kev faib tawm binomial muaj txiaj ntsig zoo hauv ntau qhov chaw, xws li kev tsim khoom, kev tshawb fawb kho mob, thiab kev suav txog kev sib raug zoo hauv zej zog, qhov twg cov txiaj ntsig ntawm kev sim rov ua dua nrog ob qhov tshwm sim tuaj yeem raug tshuaj xyuas los pab txiav txim siab. Vam tias, cov teeb meem piv txwv thiab kev sib tham tau muab yuav pab koj nkag siab ntxiv txog kev faib tawm binomial.
