Variabel acak: Béda antarrépisi

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Baris ka-16:
=== Fungsi variabel random ===
 
Lamun urang ngabogaan variabel random ''X'' on &Omega; jeung hiji [[measurable function|fungsi ukuran]] ''f'':'''R'''->'''R''', maka ''Y''=''f''(''X'') oge jadi variabel random dina &Omega;, sincesalila thefungsi compositionkomposisi ofukuran measurablebisa functions is measurablediukur. TheSababaraha sameprosedur procedurengijinkeun thatkeur allowedngarobah onetina toruang go from a probability spaceprobabiliti (&Omega;,P) tokana ('''R''',dF<sub>''X''</sub>) can be used tobisa obtaindipake thekeur probabilitynangtukeun distributionsebaran ofprobabiliti ''Y''.
Fungsi sebaran kumulatif ''Y'' nyaeta
The cumulative distribution function of ''Y'' is
 
:<math>F_Y(y) = \operatorname{P}(f(X) < y).</math>
Baris ka-23:
==== Conto ====
 
LetAnggap ''X'' be a realnilai-valuedriil randomvariabel variableacak andsarta letanggap ''Y'' = ''X''<sup>2</sup>. ThenMangka,
 
:<math>F_Y(y) = \operatorname{P}(X^2 < y).</math>
 
IfLamun ''y'' < 0, thenmangka P(''X''<sup>2</sup> &le; ''y'') = 0, somangka
 
:<math>F_Y(y) = 0\qquad\hbox{if}\quad y < 0.</math>
 
IfLamun ''y'' &ge; 0, thenmangka
 
:<math>\operatorname{P}(X^2 < y) = \operatorname{P}(|X| < \sqrt{y})
= \operatorname{P}(-\sqrt{y} < X < \sqrt{y}),</math>
 
mangka
so
 
:<math>F_Y(y) = F_X(\sqrt{y}) - F_X(-\sqrt{y})\qquad\hbox{if}\quad y \ge 0.</math>