Independence (probability theory)

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In probability theory, two events are independent , independent statistically , or stochastically independent if one does not affect the probability of another occurrence. Similarly, two random variables are independent if one realization does not affect the other probability distribution.

The concept of freedom extends to deal with the collection of more than two events or random variables, in which case the events pair independently if each pair is independent of each other, and the event is independent each if each event is independent of each combination of other events.

Video Independence (probability theory)

Definitions

For event

Two events

Dua peristiwa A dan B adalah mandiri (sering ditulis sebagai ${\ displaystyle A \ pel B}  ÂÂ$ atau ${\ displaystyle A \ pelet \! \! \! \ perp B}  ÂÂ$ ) jika dan hanya jika probabilitas gabungannya sama dengan produk dari probabilitasnya:

${\ displaystyle \ mathrm {P} (A \ cap B) = \ mathrm {P} (A) \ mathrm {P} (B)}  ÂÂ$ .

Mengapa ini mendefinisikan independensi dibuat jelas dengan menulis ulang dengan probabilitas bersyarat:

${\ displaystyle \ mathrm {P} (A \ cap B) = \ mathrm {P} (A) \ mathrm {P} (B) \ Kirirightarrow \ mathrm {P} (A) = {\ frac {\ mathrm {P} (A \ cap B)} {\ mathrm {P} (B)}} = \ mathrm {P} (A \ mid B)}  ÂÂ$ .

dan juga

${\ displaystyle \ mathrm {P} (A \ cap B) = \ mathrm {P} (A) \ mathrm {P} (B) \ Kirirightarrow \ mathrm {P} (B) = \ mathrm {P} (B \ mid A)}  ÂÂ$ .

Thus, the occurrence of B does not affect the probability of A , and vice versa. Although derived expressions may seem more intuitive, they are not a preferred definition, since conditional probabilities may not be defined if P ( A ) or P (< i> B ) is 0. Furthermore, the preferred definition makes it clear with the symmetry that when A is not dependent on B , B also does not depend on A .

More than two occasions

Kumpulan peristiwa terbatas ${\ displaystyle \ {A_ {i} \} _ {i = 1} ^ {n}}  ÂÂ$ adalah berpasangan yang berpasangan jika setiap pasangan peristiwa bersifat independen - yaitu, jika dan hanya jika untuk semua pasangan indeks yang berbeda m , k ,

${\ displaystyle \ mathrm {P} (A_ {m} \ topi A_ {k}) = \ mathrm {P} (A_ {m}) \ mathrm {P} ( A_ {k}).}  ÂÂ$

Kumpulan peristiwa terbatas adalah saling independen jika setiap peristiwa bersifat independen dari setiap persimpangan dari peristiwa lain - yaitu, jika dan hanya jika untuk setiap k -setemen subset dari ${\ displaystyle \ {A_ {i} \} _ {i = 1} ^ {n}}  ÂÂ$ ,

${\ displaystyle \ mathrm {P} \ left (\ bigcap _ {i = 1} ^ {k} A_ {i} \ right) = \ prod _ {i = 1 } ^ {k} \ mathrm {P} (A_ {i}).}  ÂÂ$

This is called multiplication rules for independent events. Note that it is not a single condition that only involves the product of all probabilities of all single events (see below for the wrong example); it should apply to all subsets of events.

For more than two events, a series of mutually independent events are (by definition) in pairs independently; but otherwise not always true (see below for a counterexample).

For real random variable

Two random variables

Two random variables X and Y are independent if and only if (iff) the elements of the system generated by them are independent; that is, for every a and b , events { X <= /i> <= b } is an independent event (as defined above). That is, X and Y with the cumulative distribution function ${\ displaystyle F_ {X} (x)}$ and ${\ displaystyle F_ {Y} (y)}$ , and the probability density ${\ displaystyle f_ {X} (x)}$ and ${\ displaystyle f_ {Y} (y)}$ X , Y ) has a cumulative distribution function together $xmlns\; =\; "http://www.w3.org/1998/Math/MathML"\; alttext\; =\; "\{\backslash \; displaystyle\; F\_\; \{X,\; Y\}\; (x,\; y)\; =\; F\_\; \{X\}\; (x)\; F\_\; \{Y\}\; (y),\}\; ">\; Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂF_{Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂX}}$,             Y                   ()         x ,         y         )         =                   F                      X                   ()         x         )                   F                      Y                   ()         y         ) ,           (X, y) = F_ {X} (x) F_ {Y} (y),} {\ displaystyle F_ {X, Y} >  Â

atau ekuivalen, jika kepadatan sendi ada,

${\ displaystyle f_ {X, Y} (x, y) = f_ {X} (x) f_ {Y} (y).}  ÂÂ$

Lebih dari dua variabel acak

A set of random variables is pairwise paired if and only if each random variable pair is independent. Even if the set of random variables are pairwise independent, they are not necessarily mutually independent as defined subsequently.

Satu set variabel acak saling independen jika dan hanya jika untuk setiap subset terbatas ${\ displaystyle X_ {1}, \ ldots, X_ {n}}  ÂÂ$ dan urutan angka yang terbatas ${\ displaystyle a_ {1}, \ ldots, a_ {n}}  ÂÂ$ , peristiwa ${\ displaystyle \ {X_ {1} \ leq a_ {1} \}, \ ldots, \ {X_ {n} \ leq a_ {n} \}}  ÂÂ$ adalah peristiwa yang saling independen (sebagaimana didefinisikan di atas).

The theoretical sizes tend to choose to change the event { X ? A } for the event { X <= a } in the above definition, where A is what Borel bundle even. The definition is exactly the same as the one above when the values ​​of the random variable are real numbers. It has the advantage of working also for random variables that are complex values ​​or for random variables taking values ​​in a scalable space (which includes a topological space endowed by? -algebras).

Conditional autonomy

Intuitively, two random variables X and Y are independently conditional given Z if, after Z is known, the value of Y did not add any additional information about X . For example, two measurements of X and Y of the same underlying quantity Z are not independent, but they are freely conditional given Z (unless the error in both measurements is somehow connected).

Definisi formal kemandirian bersyarat didasarkan pada gagasan distribusi bersyarat. Jika X , Y , dan Z adalah variabel acak diskrit, maka kita mendefinisikan X dan Y menjadi bebas kondisional diberikan Z jika

${\ displaystyle \ mathrm {P} (X \ leq x, Y \ leq y \; | \; Z = z) = \ mathrm {P} (X \ leq x \; | \; Z = z) \ cdot \ mathrm {P} (Y \ leq y \; | \; Z = z)}  ÂÂ$

untuk semua x , y dan z sehingga P ( Z Â = Â z ) Â & gt; Â 0. Di sisi lain, jika variabel acak kontinu dan memiliki fungsi kepadatan probabilitas gabungan p , maka X dan Y secara kondisional independen diberikan Z jika

${\ displaystyle p_ {XY | Z} (x, y | z) = p_ {X | Z} (x | z) \ cdot p_ {Y | Z} (y | z)}  ÂÂ$

for all real numbers i y and z until p _Z ( z ) Ã, & gt; Ã, 0.

Jika diskrit X dan Y diberikan secara independen bebas Z , maka

${\ displaystyle \ mathrm {P} (X = x | Y = y, Z = z) = \ mathrm {P} (X = x | Z = z)}  ÂÂ$

for any x , y and z with P ( Z ) Ã, Â · That is, the conditional distribution for X is given Y and Z is the same as given Z . Similar equations apply to conditional probability density functions in a continuous case.

Independence can be seen as a special kind of conditional independence, since probability can be seen as some sort of conditional probability that is not given an event.

Mandiri? -algebras

Demikian juga, keluarga terbatas dari -algebras ${\ displaystyle (\ tau _ {i}) _ {i \ dalam I}}  ÂÂ$ , di mana ${\ displaystyle I}  ÂÂ$ adalah kumpulan indeks, dikatakan independen jika dan hanya jika

${\ displaystyle \ forall \ left (A_ {i} \ right) _ {i \ in I} \ in \ prod \ nolimits _ {i \ in I} \ tau _ {i} \: \ \ mathrm {P} \ kiri (\ bigcap \ nolimits _ {i \ in I} A_ {i} \ right) = \ prod \ nolimits _ {i \ in I} \ mathrm {P} \ kiri (A_ {i} \ right)}  ÂÂ$

and the infinite family of - algebra is said to be independent if all its limited subfamilies are independent.

The new definition relates to the very first one:

• Two events are independent (in the old sense) if and only if the -gebras they produce are independent (in the new sense). The-algebra generated by an event ${\ displaystyle E \ dalam \ Sigma}$ is, by definition,

${\ displaystyle \ sigma (\ {E \}) = \ {\ emptyset, E, \ Omega \ setminus E, \ Omega \}.} Â Â$

• Two random variables X and Y are specified? are independent (in the old sense) if and only if the -algebra they produce is independent (in the new sense). The-algebra generated by a random variable X takes values ​​in some measured spaces S composed, by definition, of all parts of the set? of the form X ^-1 ( U ), where U S .

Using this definition, it is easy to show that if X and Y are random variables and Y is constant, then X and Y are independent, since the algebra generated by a constant random variable is a trivial thing? -algebra {?, Ã,?}. The probability of zero events can not affect independence so that independence also applies if Y is only Pr-almost certainly constant.

Maps Independence (probability theory)

Properties

Self-reliance

Perhatikan bahwa suatu peristiwa tidak bergantung pada dirinya sendiri jika dan hanya jika

${\ displaystyle \ mathrm {P} (A) = \ mathrm {P} (A \ cap A) = \ mathrm {P} (A) \ cdot \ mathrm {P} } (A) \ Kirirightarrow \ mathrm {P} (A) = 0 {\ text {or}} 1}  ÂÂ$ .

So an event is independent of itself if and only if it is almost certain or the extension is almost certain; This fact is useful when proving zero-one law.

Expectations and covariates

Jika X dan Y mandiri, maka operator harapan E memiliki properti

${\ displaystyle E [XY] = E [X] E [Y],}  ÂÂ$

dan covariance cov ( X , Y ) adalah nol, karena kita memiliki

$                                    cov                   [          X         ,          Y         ]          =          E          [          X          Y         ]          -          E          [          X         ] Source of the article : Wikipedia Langganan: Posting Komentar (Atom) Best Deal Today Most Visited Malaysian identity card src: cdn3.thecoverage.my Malaysian identity card (Malay: Malaysia introduction card ), is a compulsory identity card for Malaysian citi... Missouri src: img1.southernliving.timeinc.net Missouri is a state in the Midwestern United States. With over six million inhabitants, this is the 1... Sharon Kinne src: cdn.shopify.com Sharon Elizabeth Kinne , born in Sharon Elizabeth Hall , 30 November 1939), known in Mexico as La Pistolera , is an A... Synonym (taxonomy) src: i.ytimg.com In scientific nomenclature, a synonym is a scientific name that applies to a taxon that (now) goes by a different scientif... Post-9/11 Veterans Educational Assistance Act of 2008 src: iava.org The Post-9/11 Veterans Educational Assistance Act of 2008 is Title V of the Supplemental Appropriations Act of 2008, Pub.L. 1... Sponsored Links Blog Archives ▼  2018 (2336) Juli (153) Juni (258) Mei (395) April (290) Maret (133) Februari (464) Januari (643) ►  2017 (768) Desember (661) November (107) Diberdayakan oleh Blogger. Formulir Kontak Nama Email * Pesan *   About Contact Privacy Policy Sitemaps Copyleft © 2017 Depend Info $