Isibonelo Sombuzo Wengxoxo Ngenani Elilindelekile Lokusabalalisa Okuvamile
Ukusatshalaliswa okuvamile, okwaziwa nangokuthi ukusatshalaliswa kwe-Gaussian, kungenye yezindlela ezisetshenziswa kakhulu zokusatshalaliswa kwamathuba okuqhubekayo kuzibalo kanye namathuba. Lokhu kusatshalaliswa kuvame ukusetshenziswa njengesisekelo sokucabanga ezinqubweni ezahlukahlukene zezibalo ngenxa yezakhiwo zayo ezinhle zezibalo, njengokulingana kanye nokuhluka kwayo ekulinganiseni okunesilinganiso (µ) kanye nokuphambuka okujwayelekile (σ). Lesi sihloko sizoxoxa ngezibonelo futhi sixoxe ngenani elilindelekile lokusatshalaliswa okuvamile ukuze kunikezwe ukuqonda okujulile kwalomqondo.
Ukuqonda Ukusatshalaliswa Okuvamile
Ukusatshalaliswa okuvamile kuboniswa yi-bell curve ehambisanayo, enamanani amaningi agxile kunani eliphakathi, noma isilinganiso. Ngaphakathi kwalokhu kusatshalaliswa, isilinganiso (µ) kanye nokuphambuka okujwayelekile (σ) kuyimingcele emibili ebalulekile enquma indawo kanye nenani lokusabalala kwedatha.
Umsebenzi wobuningi bamathuba (PDF) wokusatshalaliswa okujwayelekile uthi:
\[f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x – \mu)^2}{2\sigma^2}}\]
Kuphi:
– \( \mu \) isilinganiso noma isilinganiso
– \( \sigma \) ukuphambuka okujwayelekile
– \( x \) iyi-variable engahleliwe
Inani Elilindelekile Ekusabalalisweni Okuvamile
Inani elilindelekile le-variable engahleliwe enokusabalalisa okuvamile lilingana nesilinganiso sokusabalalisa. Uma \( X \sim N(\mu, \sigma^2) \), khona-ke inani elilindelekile \( E(X) \) lingu:
\[ E(X) = \mu \]
Ake siqhubeke nezibonelo zezinkinga mayelana namanani alindelekile ekusatshalalisweni okuvamile ukuze siqinise ukuqonda kwethu.
Imibuzo Eyisibonelo Nengxoxo
Isibonelo Sombuzo 1:
Ake sithi \( X \) iyi-variable engahleliwe evame ukusatshalaliswa nge \( \mu = 50 \) kanye \( \sigma = 10 \). Bala inani elilindelekile le- \( X \).
Ingxoxo:
Njengoba kushiwo ngaphambili, ekusabalaleni okuvamile, inani elilindelekile \( E(X) \) lilingana no \( \mu \). Ngakho-ke,
\[ E(X) = \mu = 50 \]
Isibonelo Sombuzo 2:
Uma kunikezwe i-variable engahleliwe \( Y \) ivame ukusatshalaliswa nge \( \mu = 120 \) kanye \( \sigma = 15 \). Thola inani elilindelekile le- \( Y \).
Ingxoxo:
Ngokufanayo nesibonelo sokuqala, inani elilindelekile lika-\( Y \) liyinani eliphakathi noma isilinganiso sokusatshalaliswa okuvamile, okungukuthi:
\[ E(Y) = \mu = 120 \]
Isibonelo Sombuzo 3:
Uma i-random variable \( Z \) ilandela ukusatshalaliswa okuvamile nge \( \mu = 0 \) kanye \( \sigma = 1 \) (ukusatshalaliswa okuvamile okujwayelekile), yiliphi inani elilindelekile le \( Z \)?
Ingxoxo:
Ukusatshalaliswa okujwayelekile okujwayelekile kunesilinganiso \( \mu = 0 \), ngakho-ke inani elilindelekile \( E(Z) \) lithi:
\[ E(Z) = \mu = 0 \]
Isibonelo Sombuzo 4:
Ake sithi i-\( W \) iyi-variable engahleliwe evame ukusatshalaliswa ene-mean \( \mu = 75 \) kanye ne-standard deviation \( \sigma = 20 \). Uma sichaza i-variable entsha engahleliwe \( V = 2W + 3 \), yiliphi inani elilindelekile le-\( V \)?
Ingxoxo:
Ukuze sithole inani elilindelekile lika-\( V \), sidinga ukusebenzisa i-linearity property yenani elilindelekile. Uma sibheka \( V = 2W + 3 \), bese kuba:
\[ E(V) = E(2W + 3) \]
Ngokusekelwe kumpahla yokulingana yenani elilindelekile, singahlukanisa okungaguquki kokuguquguqukayo okungahleliwe:
\[ E(V) = 2E(W) + E(3) \]
Ukwazi ukuthi inani elilindelwe le-constant yi-constant uqobo:
\[ E(3) = 3 \]
Futhi inani elilindelekile le- \( W \) liyisilinganiso sokusatshalaliswa okuvamile \( W \):
\[ E(W) = \mu = 75 \]
Ngakho-ke,
\[ E(V) = 2 \izikhathi ezingu-75 + 3 \]
\[ E(V) = 150 + 3 \]
\[ E(V) = 153 \]
Isibonelo Sombuzo 5:
I-variable engahleliwe \( Q \) ilandela ukusatshalaliswa okuvamile okune-mean \( \mu = 40 \) kanye nokuphambuka okujwayelekile \( \sigma = 5 \). Liyini inani elilindelekile le-\( Q \) uma \[ U = Q/2 \]?
Ingxoxo:
Sisebenzisa isimiso esifanayo njengasesibonelweni sesi-4, okungukuthi i-linearity property yenani elilindelekile. Njengoba kunikezwe ukuthi \( U = Q/2 \), khona-ke:
\[ E(U) = E\kwesobunxele(\frac{Q}{2}\kwesokudla) \]
Ngokusekelwe kumpahla yokulingana yenani elilindelekile:
\[ E(U) = \frac{1}{2} E(Q) \]
Siyazi ukuthi inani elilindelekile lika-\( Q \) liyisilinganiso sokusatshalaliswa okuvamile \( Q \):
\[ E(Q) = \mu = 40 \]
Ngakho-ke,
\[ E(U) = \frac{1}{2} \izikhathi ezingu-40 \]
\[ E(U) = 20 \]
Isiphetho
Ekusabalalisweni okuvamile, inani elilindelwe le-variable engahleliwe lihlala lilingana ne-mean (µ) yokusabalalisa. Izinkinga zesibonelo ezingenhla zibonisa izimo ezahlukahlukene zokubala inani elilindelwe kusetshenziswa i-linearity property. Ukuqonda lo mqondo oyisisekelo kwenza kube lula ukusingatha izinkinga zokusabalalisa ezijwayelekile kuzibalo kanye namathuba.
Ukusatshalaliswa okuvamile kubalulekile ezibalweni ngoba kusetshenziswa ezinhlobonhlobo zezicelo ezisebenzayo, okuhlanganisa ukuhlolwa kwe-hypothesis, ukulinganisa amapharamitha, kanye nezinye iziphetho ezahlukahlukene zezibalo. Ukuqonda kahle inani elilindelekile lalokhu kusatshalaliswa kuyisinyathelo sokuqala esibalulekile ekuhlaziyweni kwedatha.
Ngethemba ukuthi lesi sihloko sizonikeza incazelo ecacile newusizo yenani elilindelekile ekusatshalalisweni okuvamile kanye nemibuzo nezingxoxo ezifanele.
