Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o ka hoʻokaʻawale ʻana o ka hiki
ʻO ka hoʻokaʻawale ʻana o ka hiki ke hana ʻia kekahi o nā manaʻo nui i ka helu helu a me ka hiki ke hana. Hoʻohana ʻia ia e hoʻomaopopo i ka hiki ke hana ʻia o nā waiwai like ʻole o kahi helu kaulele. Hiki i nā hoʻokaʻawale hiki ke hana i nā ʻano he nui ma muli o ke ʻano o ka ʻikepili e kālailai ʻia nei. ʻO nā ʻano ʻelua maʻamau o nā hoʻokaʻawale hiki ke hana ʻia he discrete a hoʻomau. Ma kēia ʻatikala, e nānā mākou i kekahi mau pilikia hoʻohālike a kūkākūkā i nā hoʻokaʻawale hiki ke hana ʻia e kōkua iā mākou e hoʻomaopopo maikaʻi i kēia kumuhana.
Ka Hoʻolaha Kūʻokoʻa
ʻO ka hoʻokaʻawale ʻokoʻa kahi hoʻokaʻawale e helu ana i ka hiki ke loaʻa kahi loli random discrete, ʻo ia hoʻi, kahi loli hiki ke lawe wale i kekahi mau waiwai. ʻO nā hiʻohiʻona kaulana o nā hoʻokaʻawale ʻokoʻa ʻo ia ka Binomial Distribution a me ka Poisson Distribution.
Laʻana 1: Hoʻokaʻawale Binomial
Hōʻike ka hoʻokaʻawale binomial i ka helu o nā holomua i loko o kahi moʻo o nā hoʻokolohua Bernoulli. Loaʻa i kēlā me kēia hoʻokolohua Bernoulli ʻelua mau hopena: holomua a hāʻule paha. Noho mau ka likelika o ka holomua i loko o ka hoʻokolohua.
Nīnau:
Ke hoʻāʻo nei kahi ʻoihana lāʻau lapaʻau i kahi lāʻau hou ma luna o 10 mau maʻi. ʻO ka likelika e hana ai ka lāʻau i loko o kekahi maʻi he 0.7. E helu i ka likelika e hana ai ka lāʻau i loko o 7 mai loko mai o 10 mau maʻi.
Kūkākūkā:
Ua hahai ka loli kaulele \(X\) i kahi hoʻolaha binomial me \(n = 10\) a me \(p = 0.7\). ʻO ka hana hikiwawe binomial:
\[ P(X = k) = \binom{n}{k} p^k (1 – p)^{n – k} \]
No \(k = 7\):
\[ P(X = 7) = \binom{10}{7} (0.7)^7 (0.3)^3 \]
Ke helu nei i ka helu binomial \(\binom{10}{7}\):
\[ \binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!} = 120 \]
Ke helu nei i nā waiwai kūpono:
P(X = 7) = 120 manawa (0.7)^7 manawa (0.3)^3
P(X = 7) kokoke i 120 manawa 0.0823543 manawa 0.027
\[ P(X = 7) \approx 0.231 \]
No laila, ʻo ka likelika e hana ai ka lāʻau lapaʻau ma 7 o 10 mau maʻi ma kahi o 0.231 a i ʻole 23.1%.
Laʻana 2: Hoʻokaʻawale Poisson
Hoʻohana ʻia ka hoʻokaʻawale Poisson e hoʻohālike i ka helu o nā hanana o kahi hanana laha ʻole i loko o kahi manawa i hāʻawi ʻia a i ʻole kahi ākea.
Nīnau:
Loaʻa i kahi hale kūʻai he awelika o 4 mau mea kūʻai i kēlā me kēia hola. He aha ka likelika e loaʻa i ka hale kūʻai he 5 mau mea kūʻai pololei i hoʻokahi hola?
Kūkākūkā:
Ua hahai ke loli kaulele \(X\) i kahi hoʻolaha Poisson me ka palamita \(\lambda = 4\). ʻO ka hana nuipa Poisson probability penei:
\[ P(X = k) = \frac{\lambda^ke^{-\lambda}}{k!} \]
No \(k = 5\):
\[ P(X = 5) = \frac{4^5 e^{-4}}{5!} \]
Helu:
\[ P(X = 5) = \frac{1024 \cdot e^{-4}}{120} \]
\[ P(X = 5) \approx \frac{1024 \cdot 0.0183}{120} \]
\[ P(X = 5) \approx 0.156 \]
No laila, ʻo ka likelika e loaʻa i ka hale kūʻai he 5 mau mea kūʻai aku i hoʻokahi hola ma kahi o 0.156, a i ʻole 15.6%.
Hoʻolaha mau
Hoʻohana ʻia nā hoʻolaha hoʻomau i ka wā e hiki ai i ke loli kaulele e ana ʻia ke lawe i kekahi waiwai i loko o kahi laulā. ʻO nā hiʻohiʻona kaulana o nā hoʻolaha hoʻomau ʻo ia ka Mahele Maʻamau a me ka Mahele Exponential.
Laʻana 3: Ka Hoʻolaha Maʻamau
ʻO ka Mahele Maʻamau, i kapa pinepine ʻia ʻo ka Mahele Gaussian, he mahele i hoʻohana pinepine ʻia ma nā ʻano like ʻole, me ka ʻepekema, ka ʻenekinia, a me ka hoʻokele waiwai.
Nīnau:
Ua puʻunaue maʻamau ʻia ke kiʻekiʻe o nā kāne makua ma ke kūlanakauhale me ka awelika o 170 kenimika a me ka ʻokoʻa maʻamau o 10 kenimika. He aha ka likelika e hiki ai i ke kāne i koho ʻia ma ka random ma waena o 160 kenimika a me 180 kenimika ke kiʻekiʻe?
Kūkākūkā:
Pono mākou e helu i ka helu-z no 160 kenimika a me 180 kenimika. Ua wehewehe ʻia ka helu-z penei:
\[ Z = \frac{X – \mu}{\sigma} \]
No \(X = 160\):
\[ Z_{160} = \frac{160 – 170}{10} = -1 \]
No \(X = 180\):
\[ Z_{180} = \frac{180 – 170}{10} = 1 \]
I kēia manawa, pono mākou e nānā i nā waiwai likelika mai -1 a i 1 ma ka papa z. ʻO ka waiwai mai z = -1 a i z = 1 ma kahi o 0.6826.
No laila, ʻo ka likelika e kū ana ke kanaka i koho ʻia ma ka hewa ma waena o 160 kenimika a me 180 kenimika ke kiʻekiʻe ma kahi o 0.6826 a i ʻole 68.26%.
Laʻana 4: Hoʻolaha Exponential
Hoʻohana ʻia ka Hoʻokaʻawale Exponential e hoʻohālike i ka manawa ma waena o nā hanana i loko o ke kaʻina hana Poisson.
Nīnau:
ʻO ka awelika o ka manawa ma waena o ka hōʻea ʻana o ʻelua mau mea kūʻai aku ma kahi hale kūʻai he 15 mau minuke. He aha ka likelika e emi iho ka manawa ma waena o ka hōʻea ʻana o ʻelua mau mea kūʻai aku ma mua o 10 mau minuke?
Kūkākūkā:
Loaʻa i ka Exponential Distribution kahi palena \(\lambda\) ʻo ia ka inverse o ka mean (\(\mu\)). Me ka mean o 15 mau minuke:
\[ \lambda = \frac{1}{\mu} = \frac{1}{15} = 0.0667 \]
ʻO ka hana hoʻolaha hōʻuluʻulu exponential:
P(X \leq x) = 1 – e^{-\lambda x} \]
No \(x = 10\):
P(X \leq 10) = 1 – e^{-0.0667 \times 10}
\[ P(X \leq 10) = 1 – e^{-0.667} \]
P(X \leq 10) \approx 1 – 0.5134 \]
\[ P(X \leq 10) \approx 0.4866 \]
No laila, ʻo ka likelika e emi iho ka manawa ma waena o nā hōʻea ʻana o nā mea kūʻai aku ʻelua ma mua o 10 mau minuke ma kahi o 0.4866 a i ʻole 48.66%.
Ka hopena
ʻO nā hoʻolaha likelike, ʻo ia hoʻi ka discrete a me ka hoʻomau, he mau manaʻo pono loa ia no ke kumu hoʻohālike a me ka hoʻomaopopo ʻana i ke ʻano o nā loli random. Hoʻohana pinepine ʻia nā hoʻolaha binomial a me Poisson no nā loli discrete, ʻoiai ʻo nā hoʻolaha maʻamau a me ka exponential he mau laʻana o nā hoʻolaha hoʻomau.
Ma o nā hiʻohiʻona ma luna, ke lana nei ko mākou manaʻo ua loaʻa iā ʻoe kahi ʻike maikaʻi aʻe i ke ʻano o ka helu ʻana a me ka wehewehe ʻana i nā probabilities i nā distributions probability. Me ka hoʻomaʻamaʻa mau ʻana, e hoʻomaikaʻi ʻia kou hiki ke hoʻomaopopo i nā distributions probability a hiki ke hoʻohana ʻia ma nā ʻano aʻo like ʻole.
