Ajụjụ Ihe Nlereanya Na-ekwu Maka Ọrụ Nkesa Binomial
Nkesa binomial bụ nkesa puru omume dị iche iche nke na-akọwa ọnụọgụ ihe ịga nke ọma na nnwale nke nwere ọtụtụ nnwale onwe ha nwere ihe abụọ nwere ike ịpụta: ihe ịga nke ọma na ọdịda. A na-akpọ nnwale ọ bụla nnwale, a na-ejikarị nkesa binomial eme ihe n'ọnọdụ ebe ọnụọgụ ihe ịga nke ọma n'ọtụtụ nnwale onwe ha dị mkpa. N'isiokwu a, anyị ga-atụle echiche ndị bụ isi nke nkesa binomial ma nye ihe atụ na ngwọta.
Echiche Ndị Dị Mkpa nke Ọrụ Nkesa Binomial
Tupu anyị abanye n'ajụjụ na mkparịta ụka ihe atụ, ka anyị tụlee ụfọdụ echiche bụ isi metụtara nkesa binomial.
1. Nkọwa: A na-akọwa nkesa binomial dị ka ngụkọta nke ihe ịga nke ọma na nnwale 'n' onwe, ebe nnwale ọ bụla nwere ihe abụọ nwere ike ịpụta: ihe ịga nke ọma (na ihe nwere ike ime p) ma ọ bụ ọdịda (na ihe nwere ike ime q = 1 – p).
2. Ọrụ Ohere: Ọrụ puru omume nke nkesa binomial bụ:
\[
P(X = k) = \binom{n}{k} p^k (1-p)^{nk}
\]
Ebe:
– \( P(X = k) \) bụ ohere nke inwe ihe ịga nke ọma k na nnwale n.
– \( \binom{n}{k} \) bụ njikọta nke n were k, nke a kọwara dị ka \( \frac{n!}{k!(nk)!} \).
– \( p \) bụ ohere nke ihe ịga nke ọma na nnwale ọ bụla.
– \( (1-p) \) bụ ohere nke ọdịda na nnwale ọ bụla.
3. Uru a tụrụ anya ya na mgbanwe dị iche iche:
– Uru a tụrụ anya ya (nkezi) nke nkesa binomial bụ \( \mu = np \).
– Mgbanwe nke nkesa binomial bụ \( \sigma^2 = np(1-p) \).
Ugbua, ka anyị tinye echiche ndị a n'ọrụ n'ihe atụ iji nye nghọta miri emi.
Ajụjụ Ihe Nlereanya nke 1: Ngụkọta Isi nke Nkesa Binomial
Ajụjụ:
Ụlọ ọrụ na-emepụta ihe eletrọnịkị nwere ohere 0.95 na ihe ọ bụla gafere ule mma. Ọ bụrụ na emepụta ihe iri, gbakọọ ohere na ihe asatọ kpọmkwem gafere ule mma.
Mkparịta ụka:
Anyị nwere ike iji usoro nkesa binomial dozie nsogbu a. Nke mbụ, anyị na-achọpụta paramita ndị a:
– \( n \) (ọnụọgụ nnwale niile) = 10
– \( k \) (ọnụọgụ ihe ịga nke ọma) = 8
– \( p \) (ihe ga-eme ka ihe gaa nke ọma) = 0.95
– \( q \) (ihe nwere ike ime ka ọdịda daa) = 1 – 0.95 = 0.05
Mgbe ahụ, tinye ụkpụrụ ndị a n'ime usoro nkesa binomial:
\[
P(X = 8) = \binom{10}{8} (0.95)^8 (0.05)^2
\]
Nke mbụ, gbakọọ njikọta \( \binom{10}{8} \):
\[
\binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10!}{8!2!} = \frac{10 \u003d 9 \u003d 8!}{8! \u003d 2!} = \frac{10 \u003d 9}{2 \u003d 1} = 45
\]
Mgbe ahụ, gbakọọ ihe gbasara ohere \( (0.95)^8 \) na \( (0.05)^2 \):
\[
(0.95)^8 \ihe dị ka 0.6634
\]
\[
(0.05)^2 = 0.0025
\]
N'ikpeazụ, mụbaa ụkpụrụ ndị ahụ niile iji nweta:
\[
P(X = 8) = 45 \ugboro 0.6634 \ugboro 0.0025 \uihe dị ka 0.0744
\]
Ya mere, ohere na kpọmkwem ihe 8 n'ime ihe 10 gafere ule mma bụ ihe dịka 0.0744 ma ọ bụ 7.44%.
Ajụjụ Ihe Nlereanya nke 2: Ngụkọta Ihe Nwere Ike Ime
Ajụjụ:
Ka ọ dị n'otu ụlọ ọrụ ahụ, gbakọọ ohere na ihe ruru 9 n'ime 10 ga-agafe ule mma ahụ.
Mkparịta ụka:
Iji dozie nsogbu a, anyị kwesịrị ịgbakọ ohere mkpokọta. Ohere nke ihe ruru 9 n'ime 10 gafere ule ahụ pụtara na anyị gbakọọ \( P(X \geq 9) \), nke enwere ike ide dị ka:
\[
P (X \geq 9) = P (X = 9) + P (X = 10)
\]
Site na iji usoro nkesa binomial:
\[
P(X = 9) = \binom{10}{9} (0.95)^9 (0.05)^1
\]
\[
P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^0
\]
Nke mbụ, gbakọọ njikọta maka ikpe ọ bụla:
\[
\binom{10}{9} = \frac{10!}{9!(10-9)!} = 10
\]
\[
\binom{10}{10} = 1
\]
Mgbe ahụ, gbakọọ ihe gbasara ohere maka \( P(X = 9) \) na \( P(X = 10) \):
\[
P(X = 9) = 10 \ugboro (0.95)^9 \ugboro 0.05
\]
\[
(0.95)^9 \ihe dị ka 0.6302
\]
\[
P(X = 9) = 10 \ugboro 0.6302 \ugboro 0.05 \uihe dị ka 0.3151
\]
\[
P(X = 10) = 1 \ugboro (0.95)^{10} \ugboro 1
\]
\[
(0.95)^{10} \ihe dị ka 0.5987
\]
\[
P(X = 10) = 0.5987
\]
Ngụkọta ohere maka \( P(X \geq 9) \):
\[
P(X \geq 9) = 0.3151 + 0.5987 \ihe dị ka 0.9138
\]
Ya mere, ohere na ihe ruru 9 n'ime 10 gafere ule mma ahụ bụ ihe dịka 0.9138 ma ọ bụ 91.38%.
Ajụjụ Ihe Nlereanya nke 3: Uru A Na-atụ Anya Ya na Mgbanwe Ya
Ajụjụ:
Gbakọọ uru a tụrụ anya ya na ọdịiche nke ọnụọgụgụ ihe ndị gafere ule mma n'ime ihe iri e mepụtara, yana ohere nke ịgafe 0.95.
Mkparịta ụka:
Jiri usoro a:
– Uru a tụrụ anya ya (nkezi) \( \mu = np \)
– Mgbanwe \( \sigma^2 = np(1-p) \)
Na \( n = 10 \) na \( p = 0.95 \):
\[
\mu = ugboro iri 0.95 = 9.5
\]
\[
\sigma^2 = 10 \ugboro 0.95 \ugboro 0.05 = 0.475
\]
Ya mere, uru a tụrụ anya ya nke ọnụọgụgụ ihe ndị gafere ule mma bụ 9.5, mgbanwe ahụ bụkwa 0.475.
Mmechi
Site na nsogbu ihe atụ atọ dị n'elu, anyị atụleela otu esi agbakọ ihe gbasara ohere site na iji nkesa binomial maka ọnọdụ dị iche iche: ịgbakọ ihe gbasara ohere kpọmkwem, ohere mkpokọta, na ịgbakọ uru na mgbanwe a tụrụ anya ya. Ihe ọmụma nke nkesa binomial bara uru n'ọtụtụ ngalaba, dị ka mmepụta ihe, nyocha ahụike, na ọnụ ọgụgụ mmekọrịta mmadụ na ibe ya, ebe enwere ike inyocha nsonaazụ nke nnwale ugboro ugboro nwere nsonaazụ abụọ enwere ike iji nyere aka ime mkpebi. Olileanya, nsogbu ihe atụ na mkparịta ụka enyere ga-enyere aka mee ka nghọta gị banyere nkesa binomial dịkwuo mma.
