Tauira o tētahi Pātai Kōrero mō te Uara e Tūmanakohia ana o tētahi Tohatoha Noa
Ko te tohatoha noa, e mōhiotia ana ko te tohatoha Gaussian, tētahi o ngā tohatoha tūponotanga tonu e whakamahia whānuitia ana i roto i ngā tatauranga me te tūponotanga. He maha ngā wā ka whakamahia tēnei tohatoha hei whakaaro taketake i roto i ngā whakatau tatauranga maha nā ōna āhuatanga pāngarau pai, pērā i te ōritetanga me tōna ahurei i roto i te whakapūtanga tawhā me te toharite (µ) me te paerewa rerekētanga (σ). Ka matapakihia e tēnei tuhinga ngā tauira me te matapaki i te uara e tumanakohia ana o te tohatoha noa hei whakarato i te māramatanga hohonu ake mō tēnei ariā.
Te Mārama ki te Tohatoha Noa
Ko te tohatoha noa e whakaaturia ana e te pihi pere ōrite, me te nuinga o ngā uara e aro ana ki te uara waenga, te toharite rānei. I roto i tēnei tohatoha, ko te toharite (µ) me te paerewa rerekētanga (σ) he tawhā nui e rua e whakatau ana i te taunga me te nui o te horapa i roto i ngā raraunga.
Ko te mahi tūponotanga kiato (PDF) o te tohatoha noa ko:
\[f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x – \mu)^2}{2\sigma^2}}\]
kāore i te mana:
– Ko te toharite, te toharite rānei te \( \mu \)
– Ko te paerewa rerekētanga ko \( \sigma \)
– He taurangi matapōkere a \( x \)
Uara e Tumanakohia ana i roto i te Tohatoha Noa
Ko te uara e tumanakohia ana o tētahi taurangi matapōkere me te tohatoha noa he ōrite ki te toharite o te tohatoha. Mena ko \( X \sim N(\mu, \sigma^2) \), ko te uara e tumanakohia ana \( E(X) \) ko:
\[ E(X) = \mu \]
Me haere tonu tātou me ētahi tauira o ngā raruraru e pā ana ki ngā uara e tumanakohia ana i roto i ngā tohatoha noa hei whakapakari i tō tātou māramatanga.
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1:
Me kī he taurangi matapōkere tohatoha noa a \( X \) me \( \mu = 50 \) me \( \sigma = 10 \). Tātaihia te uara e tumanakohia ana o \( X \).
Kōrero:
E ai ki te kōrero i mua ake nei, i roto i te tohatoha noa, ko te uara e tumanakohia ana \( E(X) \) he ōrite ki \( \mu \). Nō reira,
\[ E(X) = \mu = 50 \]
Tauira Pātai 2:
Ka tohatoha noa te taurangi matapōkere \( Y \) me \( \mu = 120 \) me \( \sigma = 15 \). Kimihia te uara e tumanakohia ana mō \( Y \).
Kōrero:
Pērā i te tauira tuatahi, ko te uara e tumanakohia ana o \( Y \) ko te uara waenga, ko te toharite rānei o te tohatoha noa, arā:
\[ E(Y) = \mu = 120 \]
Tauira Pātai 3:
Mena ka whai te taurangi matapōkere \( Z \) i tētahi tohatoha noa me \( \mu = 0 \) me \( \sigma = 1 \) (tohatoha noa paerewa), he aha te uara e tumanakohia ana mō \( Z \)?
Kōrero:
Ko te toharite o te tohatoha noa paerewa he \( \mu = 0 \), nō reira ko te uara e tumanakohia ana \( E(Z) \) ko:
\[ E(Z) = \mu = 0 \]
Tauira Pātai 4:
Me kī ko \( W \) he taurangi matapōkere tohatoha noa me te toharite \( \mu = 75 \) me te paerewa rerekētanga \( \sigma = 20 \). Mēnā ka tautuhia e tātou he taurangi matapōkere hou \( V = 2W + 3 \), he aha te uara e tumanakohia ana o \( V \)?
Kōrero:
Hei kimi i te uara e tumanakohia ana o \( V \), me whakamahi tātou i te āhuatanga rārangi o te uara e tumanakohia ana. Mēnā ka hoatu \( V = 2W + 3 \), kātahi:
\[ E(V) = E(2W + 3) \]
I runga i te āhuatanga rārangi o te uara e tumanakohia ana, ka taea e tātou te wehewehe i te pūmau mai i te taurangi matapōkere:
\[ E(V) = 2E(W) + E(3) \]
Te mōhio ko te uara e tumanakohia ana o tētahi pūmau ko te pūmau tonu:
\[ E(3) = 3 \]
Ā, ko te uara e tumanakohia ana o \( W \) ko te toharite o te tohatoha noa \( W \):
\[ E(W) = \mu = 75 \]
Nō reira,
\[ E(V) = 2 \whakanuia te 75 + 3 \]
\[ E(V) = 150 + 3 \]
\[ E(V) = 153 \]
Tauira Pātai 5:
Ko te taurangi matapōkere \( Q \) e whai ana i te tohatoha noa me te toharite \( \mu = 40 \) me te paerewa rerekētanga \( \sigma = 5 \). He aha te uara e tumanakohia ana mō \( Q \) mēnā \[ U = Q/2 \]?
Kōrero:
Ka whakamahia e mātou te kaupapa kotahi i te tauira 4, arā, ko te āhuatanga rārangi o te uara e tumanakohia ana. I te mea ko \( U = Q/2 \), kāti:
\[ E(U) = E\maui(\frac{Q}{2}\matau) \]
I runga i te āhuatanga rārangi o te uara e tumanakohia ana:
\[ E(U) = \frac{1}{2} E(Q) \]
E mōhio ana tātou ko te uara e tumanakohia ana o \( Q \) ko te toharite o te tohatoha noa \( Q \):
\[ E(Q) = \mu = 40 \]
Nō reira,
\[ E(U) = \frac{1}{2} \times 40 \]
\[ E(U) = 20 \]
Whakamutunga
I roto i te tohatoha noa, ko te uara e tumanakohia ana o te taurangi matapōkere he rite tonu ki te toharite (µ) o te tohatoha. Ko ngā tauira raruraru i runga ake nei e whakaatu ana i ngā tikanga maha mō te tatau i te uara e tumanakohia ana mā te whakamahi i te āhuatanga raina. Mā te mārama ki tēnei ariā taketake ka māmā ake te whakahaere i ngā raruraru tohatoha noa i roto i ngā tatauranga me te tūponotanga.
He mea nui te tohatoha noa i roto i ngā tatauranga nā te mea e whakamahia ana i roto i te whānuitanga o ngā tono mahi, tae atu ki te whakamātautau whakapae, te whakatau i ngā tawhā, me ngā momo whakatau tatauranga. Ko te mārama pai ki te uara e tumanakohia ana o tēnei tohatoha he taahiraa tuatahi nui ki te tātari raraunga.
Ko te tumanako ka whakaratohia e tēnei tuhinga he whakamārama mārama, he whakamārama whai hua hoki mō te uara e tumanakohia ana i roto i te tohatoha noa, me ngā tauira pātai me ngā kōrero e tika ana.
