Laʻana o kahi Nīnau Kūkākūkā e pili ana i ka Waiwai i Manaʻo ʻia o kahi Hoʻolaha Maʻamau
ʻO ka hoʻolaha maʻamau, i ʻike ʻia hoʻi ʻo ka hoʻolaha Gaussian, ʻo ia kekahi o nā hoʻolaha hoʻomau i hoʻohana pinepine ʻia i nā helu helu a me ka hiki. Hoʻohana pinepine ʻia kēia hoʻolaha ma ke ʻano he kuhi kumu i nā ʻano kuhi helu like ʻole ma muli o kona mau waiwai makemakika maikaʻi, e like me ka symmetry a me kona ʻano kū hoʻokahi i ka parameterization me ka awelika (µ) a me ka deviation maʻamau (σ). E kūkākūkā kēia ʻatikala i nā laʻana a kūkākūkā i ka waiwai i manaʻo ʻia o ka hoʻolaha maʻamau e hāʻawi i kahi ʻike hohonu o kēia manaʻo.
Ke Hoʻomaopopo nei i ka Hoʻolaha Maʻamau
Ua hōʻike ʻia ka hoʻolaha maʻamau e kahi piʻo bele symmetrical, me ka hapa nui o nā waiwai i kālele ʻia a puni ka waiwai waena, a i ʻole ka awelika. I loko o kēia hoʻolaha, ʻo ka awelika (µ) a me ka deviation maʻamau (σ) ʻelua mau palena koʻikoʻi e hoʻoholo ai i kahi a me ka nui o ka laha ʻana ma ka ʻikepili.
ʻO ka hana density probability (PDF) o ka hoʻolaha maʻamau:
\[f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x – \mu)^2}{2\sigma^2}}\]
Ma hea:
– ʻO \( \mu \) ka awelika a i ʻole ka awelika
– ʻO \( \sigma \) ka ʻokoʻa maʻamau
– He loli kaulele ʻo \( x \)
Ka Waiwai i Manaʻo ʻia ma ka Hoʻolaha Maʻamau
ʻO ka waiwai i manaʻo ʻia o kahi loli kaulele me kahi hoʻolaha maʻamau ua like ia me ka awelika o ka hoʻolaha. Inā \( X \sim N(\mu, \sigma^2) \), a laila ʻo ka waiwai i manaʻo ʻia \( E(X) \) penei:
\[ E(X) = \mu \]
E hoʻomau kākou me kekahi mau hiʻohiʻona o nā pilikia e pili ana i nā waiwai i manaʻo ʻia ma nā hoʻolaha maʻamau e hoʻoikaika i ko kākou ʻike.
Nā Nīnau Laʻana a me ke Kūkākūkā
Laʻana Nīnau 1:
Manaʻo ʻia he loli maʻamau i hoʻokaʻawale ʻia ʻo \( X \) me \( \mu = 50 \) a me \( \sigma = 10 \). E helu i ka waiwai i manaʻo ʻia o \( X \).
Kūkākūkā:
E like me ka mea i ʻōlelo ʻia ma mua, ma kahi hoʻolaha maʻamau, ʻo ka waiwai i manaʻo ʻia \( E(X) \) ua like ia me \( \mu \). No laila,
\[ E(X) = \mu = 50 \]
Laʻana Nīnau 2:
Hāʻawi ʻia kahi loli kaulele \( Y \) ua puʻunaue maʻamau ʻia me \( \mu = 120 \) a me \( \sigma = 15 \). E huli i ka waiwai i manaʻo ʻia o \( Y \).
Kūkākūkā:
E like me ka laʻana mua, ʻo ka waiwai i manaʻo ʻia o \( Y \) ʻo ia ka waiwai waena a i ʻole ka awelika o ka hoʻolaha maʻamau, ʻo ia hoʻi:
\[ E(Y) = \mu = 120 \]
Laʻana Nīnau 3:
Inā hahai ka loli kaulele \( Z \) i kahi hoʻolaha maʻamau me \( \mu = 0 \) a me \( \sigma = 1 \) (hoʻolaha maʻamau maʻamau), he aha ka waiwai i manaʻo ʻia o \( Z \)?
Kūkākūkā:
ʻO ka awelika o ka hoʻolaha maʻamau maʻamau \( \mu = 0 \), no laila ʻo ka waiwai i manaʻo ʻia \( E(Z) \) penei:
\[ E(Z) = \mu = 0 \]
Laʻana Nīnau 4:
Manaʻo ʻia he loli maʻamau i hoʻolaha ʻia ʻo \( W \) me ka awelika \( \mu = 75 \) a me ka ʻokoʻa maʻamau \( \sigma = 20 \). Inā mākou e wehewehe i kahi loli maʻamau hou \( V = 2W + 3 \), he aha ka waiwai i manaʻo ʻia o \( V \)?
Kūkākūkā:
No ka loaʻa ʻana o ka waiwai i manaʻo ʻia o \( V \), pono mākou e hoʻohana i ka waiwai linearity o ka waiwai i manaʻo ʻia. Hāʻawi ʻia \( V = 2W + 3 \), a laila:
\[ E(V) = E(2W + 3) \]
Ma muli o ka waiwai linearity o ka waiwai i manaʻo ʻia, hiki iā mākou ke hoʻokaʻawale i ke kūpaʻa mai ka loli random:
\[ E(V) = 2E(W) + E(3) \]
ʻIke i ka waiwai i manaʻo ʻia o kahi kūpaʻa ʻo ia ke kūpaʻa ponoʻī:
\[ E(3) = 3 \]
A ʻo ka waiwai i manaʻo ʻia o \( W \) ʻo ia ka awelika o ka hoʻolaha maʻamau \( W \):
\[ E(W) = \mu = 75 \]
No laila,
\[ E(V) = 2 \times 75 + 3 \]
\[ E(V) = 150 + 3 \]
\[ E(V) = 153 \]
Laʻana Nīnau 5:
Ua hahai ka loli kaulele \( Q \) i kahi hoʻolaha maʻamau me ka awelika \( \mu = 40 \) a me ka ʻokoʻa maʻamau \( \sigma = 5 \). He aha ka waiwai i manaʻo ʻia o \( Q \) inā \[ U = Q/2 \]?
Kūkākūkā:
Hoʻohana mākou i ke kumumanaʻo like me ka laʻana 4, ʻo ia hoʻi ka waiwai linearity o ka waiwai i manaʻo ʻia. No ka mea \( U = Q/2 \), a laila:
\[ E(U) = E\left(\frac{Q}{2}\right) \]
Ma muli o ka waiwai linearity o ka waiwai i manaʻo ʻia:
\[ E(U) = \frac{1}{2} E(Q) \]
Ua ʻike mākou ʻo ka waiwai i manaʻo ʻia o \( Q \) ʻo ia ka awelika o ka hoʻolaha maʻamau \( Q \):
\[ E(Q) = \mu = 40 \]
No laila,
\[ E(U) = \frac{1}{2} \times 40 \]
\[ E(U) = 20 \]
Ka hopena
Ma kahi hoʻolaha maʻamau, ʻo ka waiwai i manaʻo ʻia o kahi loli kaulele e like mau me ka awelika (µ) o ka hoʻolaha. Hōʻike nā pilikia hoʻohālike ma luna i nā kūlana like ʻole no ka helu ʻana i ka waiwai i manaʻo ʻia me ka hoʻohana ʻana i ka waiwai linearity. ʻO ka hoʻomaopopo ʻana i kēia manaʻo kumu e maʻalahi ai ka lawelawe ʻana i nā pilikia hoʻolaha maʻamau i nā helu helu a me ka hiki.
He mea koʻikoʻi ka hoʻolaha maʻamau i nā helu helu no ka mea hoʻohana ʻia ia i nā ʻano hana like ʻole, me ka hoʻāʻo ʻana i nā kuhiakau, ka kuhi ʻana o nā palena, a me nā ʻano kuhi helu like ʻole. ʻO ka hoʻomaopopo maikaʻi ʻana i ka waiwai i manaʻo ʻia o kēia hoʻolaha he hana mua koʻikoʻi ia i ka nānā ʻana i ka ʻikepili.
Manaʻolana e hāʻawi kēia ʻatikala i kahi wehewehe maopopo a pono hoʻi o ka waiwai i manaʻo ʻia ma ka hoʻolaha maʻamau me nā nīnau hoʻohālike pili a me nā kūkākūkā.
