Ayr\u0131k bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131,<\/strong> ayr\u0131 bir rastgele de\u011fi\u015fkenin<\/a> olas\u0131l\u0131klar\u0131n\u0131 tan\u0131mlayan da\u011f\u0131l\u0131md\u0131r. Bu nedenle, ayr\u0131k bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131 yaln\u0131zca sonlu say\u0131da de\u011fer (genellikle tam say\u0131lar) alabilir.<\/p>\n \u00d6rne\u011fin binom da\u011f\u0131l\u0131m\u0131, Poisson da\u011f\u0131l\u0131m\u0131 ve hipergeometrik da\u011f\u0131l\u0131m ayr\u0131k olas\u0131l\u0131k da\u011f\u0131l\u0131mlar\u0131d\u0131r.<\/p>\n Ayr\u0131k bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131nda, (xi )<\/sub> ‘yi temsil eden ayr\u0131k de\u011fi\u015fkenin her de\u011feri, 0’dan 1’e kadar de\u011fi\u015fen bir olas\u0131l\u0131k de\u011feriyle (pi )<\/sub> ili\u015fkilendirilir. B\u00f6ylece, ayr\u0131k bir da\u011f\u0131l\u0131mdaki t\u00fcm olas\u0131l\u0131klar\u0131n toplam\u0131, bir sonucunu verir. . <\/p>\n<\/p>\n Art\u0131k ayr\u0131k olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131n\u0131n tan\u0131m\u0131n\u0131 bildi\u011fimize g\u00f6re, kavram\u0131 daha iyi anlamak i\u00e7in bu t\u00fcr da\u011f\u0131l\u0131m\u0131n birka\u00e7 \u00f6rne\u011fini g\u00f6rece\u011fiz.<\/p>\n Ayr\u0131k olas\u0131l\u0131k da\u011f\u0131l\u0131mlar\u0131na \u00f6rnekler:<\/u><\/strong><\/p>\n Ayr\u0131k olas\u0131l\u0131k da\u011f\u0131l\u0131mlar\u0131n\u0131n ana t\u00fcrleri<\/strong> \u015funlard\u0131r:<\/p>\n Ayr\u0131k olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131n\u0131n her t\u00fcr\u00fc a\u015fa\u011f\u0131da ayr\u0131nt\u0131l\u0131 olarak a\u00e7\u0131klanmaktad\u0131r. <\/p>\n Ayr\u0131k tekd\u00fcze da\u011f\u0131l\u0131m,<\/strong> t\u00fcm de\u011ferlerin e\u015f olas\u0131l\u0131kl\u0131 oldu\u011fu, yani ayr\u0131k bir tekd\u00fcze da\u011f\u0131l\u0131mda t\u00fcm de\u011ferlerin ayn\u0131 olu\u015fma olas\u0131l\u0131\u011f\u0131na sahip oldu\u011fu ayr\u0131 bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r.<\/p>\n \u00d6rne\u011fin, t\u00fcm olas\u0131 sonu\u00e7lar\u0131n (1, 2, 3, 4, 5 veya 6) ayn\u0131 olu\u015fma olas\u0131l\u0131\u011f\u0131na sahip olmas\u0131 nedeniyle, bir zar\u0131n at\u0131lmas\u0131 ayr\u0131k ve d\u00fczg\u00fcn bir da\u011f\u0131l\u0131mla tan\u0131mlanabilir.<\/p>\n Genel olarak, ayr\u0131k bir d\u00fczg\u00fcn da\u011f\u0131l\u0131m, da\u011f\u0131l\u0131m\u0131n alabilece\u011fi olas\u0131 de\u011ferlerin aral\u0131\u011f\u0131n\u0131 tan\u0131mlayan a<\/em> ve b<\/em> olmak \u00fczere iki karakteristik parametreye sahiptir. Bu nedenle, bir de\u011fi\u015fken ayr\u0131k bir d\u00fczg\u00fcn da\u011f\u0131l\u0131mla tan\u0131mland\u0131\u011f\u0131nda, D\u00fczg\u00fcn(a,b)<\/em> olarak yaz\u0131l\u0131r.<\/p>\n<\/p>\n Ayr\u0131k tekd\u00fcze da\u011f\u0131l\u0131m, rastgele deneyleri tan\u0131mlamak i\u00e7in kullan\u0131labilir \u00e7\u00fcnk\u00fc t\u00fcm sonu\u00e7lar\u0131n ayn\u0131 olas\u0131l\u0131\u011fa sahip olmas\u0131, deneyin rastgele oldu\u011fu anlam\u0131na gelir. <\/p>\n \u0130kili da\u011f\u0131l\u0131m<\/strong> olarak da bilinen Bernoulli da\u011f\u0131l\u0131m\u0131<\/strong> , yaln\u0131zca iki sonuca sahip olabilen ayr\u0131 bir de\u011fi\u015fkeni temsil eden bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r: “ba\u015far\u0131” veya “ba\u015far\u0131s\u0131zl\u0131k”.<\/p>\n Bernoulli da\u011f\u0131l\u0131m\u0131nda “ba\u015far\u0131” bekledi\u011fimiz sonu\u00e7tur ve 1 de\u011ferine sahipken, “ba\u015far\u0131s\u0131zl\u0131k” sonucu beklenenin d\u0131\u015f\u0131nda bir sonu\u00e7tur ve 0 de\u011ferine sahiptir. ba\u015far\u0131\u201d p<\/em> , \u201cba\u015far\u0131s\u0131zl\u0131k\u201d sonucunun olas\u0131l\u0131\u011f\u0131 q=1-p’dir<\/em> .<\/p>\n<\/p>\n Bernoulli da\u011f\u0131l\u0131m\u0131 ismini \u0130svi\u00e7reli istatistik\u00e7i Jacob Bernoulli’den alm\u0131\u015ft\u0131r.<\/p>\n \u0130statistikte, Bernoulli da\u011f\u0131l\u0131m\u0131n\u0131n esas olarak tek bir uygulamas\u0131 vard\u0131r: yaln\u0131zca iki olas\u0131 sonucun (ba\u015far\u0131 ve ba\u015far\u0131s\u0131zl\u0131k) oldu\u011fu deneylerin olas\u0131l\u0131klar\u0131n\u0131 tan\u0131mlamak. Dolay\u0131s\u0131yla Bernoulli da\u011f\u0131l\u0131m\u0131n\u0131 kullanan bir deneye Bernoulli testi veya Bernoulli deneyi denir. <\/p>\n Binom<\/strong> da\u011f\u0131l\u0131m\u0131 olarak da adland\u0131r\u0131lan binom da\u011f\u0131l\u0131m\u0131<\/strong> , sabit bir ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131 ile bir dizi ba\u011f\u0131ms\u0131z, ikili deney ger\u00e7ekle\u015ftirirken elde edilen ba\u015far\u0131lar\u0131n say\u0131s\u0131n\u0131 sayan bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r. Ba\u015fka bir deyi\u015fle binom da\u011f\u0131l\u0131m\u0131, bir dizi Bernoulli denemesinin ba\u015far\u0131l\u0131 sonu\u00e7lar\u0131n\u0131n say\u0131s\u0131n\u0131 tan\u0131mlayan bir da\u011f\u0131l\u0131md\u0131r.<\/p>\n \u00d6rne\u011fin, bir madalyonun 25 kez tura gelme say\u0131s\u0131 binom da\u011f\u0131l\u0131m\u0131d\u0131r.<\/p>\n Genel olarak ger\u00e7ekle\u015ftirilen deneylerin toplam say\u0131s\u0131 n<\/em> parametresi ile tan\u0131mlan\u0131rken, p<\/em> her deneyin ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131d\u0131r. B\u00f6ylece, binom da\u011f\u0131l\u0131m\u0131n\u0131 takip eden bir rastgele de\u011fi\u015fken \u015fu \u015fekilde yaz\u0131l\u0131r:<\/p>\n<\/p>\n Binom da\u011f\u0131l\u0131m\u0131nda ayn\u0131 deneyin n<\/em> kez tekrarland\u0131\u011f\u0131n\u0131 ve deneylerin birbirinden ba\u011f\u0131ms\u0131z oldu\u011funu, dolay\u0131s\u0131yla her deneyin ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131n\u0131n ayn\u0131 oldu\u011funu (p)<\/em> unutmay\u0131n. <\/p>\n Poisson da\u011f\u0131l\u0131m\u0131,<\/strong> belirli bir s\u00fcre boyunca belirli say\u0131da olay\u0131n meydana gelme olas\u0131l\u0131\u011f\u0131n\u0131 tan\u0131mlayan bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r. Ba\u015fka bir deyi\u015fle Poisson da\u011f\u0131l\u0131m\u0131, bir olgunun belirli bir zaman aral\u0131\u011f\u0131nda tekrarlanma say\u0131s\u0131n\u0131 tan\u0131mlayan rastgele de\u011fi\u015fkenleri modellemek i\u00e7in kullan\u0131l\u0131r.<\/p>\n \u00d6rne\u011fin, bir telefon santralinin dakika ba\u015f\u0131na ald\u0131\u011f\u0131 \u00e7a\u011fr\u0131 say\u0131s\u0131, Poisson da\u011f\u0131l\u0131m\u0131 kullan\u0131larak tan\u0131mlanabilen ayr\u0131 bir rastgele de\u011fi\u015fkendir.<\/p>\n Poisson da\u011f\u0131l\u0131m\u0131n\u0131n, Yunanca \u03bb harfiyle temsil edilen karakteristik bir parametresi vard\u0131r ve incelenen olay\u0131n belirli bir aral\u0131kta ka\u00e7 kez meydana gelmesinin beklendi\u011fini g\u00f6sterir. <\/p>\n<\/p>\n \u00c7ok terimli da\u011f\u0131l\u0131m<\/strong> (veya \u00e7ok terimli da\u011f\u0131l\u0131m<\/strong> ), birbirini d\u0131\u015flayan birka\u00e7 olay\u0131n birka\u00e7 denemeden sonra belirli say\u0131da meydana gelme olas\u0131l\u0131\u011f\u0131n\u0131 tan\u0131mlayan bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r.<\/p>\n Yani, rastgele bir deney \u00fc\u00e7 veya daha fazla \u00f6zel olayla sonu\u00e7lanabiliyorsa ve her olay\u0131n ayr\u0131 ayr\u0131 meydana gelme olas\u0131l\u0131\u011f\u0131 biliniyorsa, \u00e7ok terimli da\u011f\u0131l\u0131m, birden fazla deney ger\u00e7ekle\u015ftirildi\u011finde belirli say\u0131da olay\u0131n meydana gelme olas\u0131l\u0131\u011f\u0131n\u0131 hesaplamak i\u00e7in kullan\u0131l\u0131r. her zaman.<\/p>\n Dolay\u0131s\u0131yla \u00e7ok terimli da\u011f\u0131l\u0131m, binom da\u011f\u0131l\u0131m\u0131n\u0131n bir genellemesidir. <\/p>\n Geometrik da\u011f\u0131l\u0131m,<\/strong> ilk ba\u015far\u0131l\u0131 sonucu elde etmek i\u00e7in gereken Bernoulli denemelerinin say\u0131s\u0131n\u0131 tan\u0131mlayan bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r. Yani, Bernoulli deneylerinden biri pozitif sonu\u00e7 elde edene kadar tekrarlanan s\u00fcre\u00e7lerin geometrik da\u011f\u0131l\u0131m modelleri.<\/p>\n \u00d6rne\u011fin bir yoldan sar\u0131 bir araba g\u00f6rene kadar ge\u00e7en araba say\u0131s\u0131 geometrik bir da\u011f\u0131l\u0131md\u0131r.<\/p>\n Bernoulli testinin iki olas\u0131 sonucu olan bir deney oldu\u011funu unutmay\u0131n: “ba\u015far\u0131” ve “ba\u015far\u0131s\u0131zl\u0131k”. Yani \u201cba\u015far\u0131\u201d olas\u0131l\u0131\u011f\u0131 p<\/em> ise, \u201cba\u015far\u0131s\u0131zl\u0131k\u201d olas\u0131l\u0131\u011f\u0131 q=1-p’dir<\/em> .<\/p>\n Bu nedenle geometrik da\u011f\u0131l\u0131m, ger\u00e7ekle\u015ftirilen t\u00fcm deneylerin ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131 olan p<\/em> parametresine ba\u011fl\u0131d\u0131r. Ayr\u0131ca p<\/em> olas\u0131l\u0131\u011f\u0131 t\u00fcm deneyler i\u00e7in ayn\u0131d\u0131r. <\/p>\n<\/p>\n Negatif binom da\u011f\u0131l\u0131m\u0131,<\/strong> belirli say\u0131da pozitif sonu\u00e7 elde etmek i\u00e7in gereken Bernoulli denemelerinin say\u0131s\u0131n\u0131 tan\u0131mlayan bir olas\u0131l\u0131k da\u011f\u0131l\u0131m\u0131d\u0131r.<\/p>\n Bu nedenle, negatif bir binom da\u011f\u0131l\u0131m\u0131n\u0131n iki karakteristik parametresi vard\u0131r: r<\/em> , istenen ba\u015far\u0131l\u0131 sonu\u00e7lar\u0131n say\u0131s\u0131d\u0131r ve p<\/em> , ger\u00e7ekle\u015ftirilen her Bernoulli deneyinin ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131d\u0131r.<\/p>\n<\/p>\n Dolay\u0131s\u0131yla negatif binom da\u011f\u0131l\u0131m\u0131, pozitif sonu\u00e7lar<\/em> elde etmek i\u00e7in gerekti\u011fi kadar Bernoulli denemesinin yap\u0131ld\u0131\u011f\u0131 bir s\u00fcreci tan\u0131mlar. Ayr\u0131ca, t\u00fcm bu Bernoulli denemeleri ba\u011f\u0131ms\u0131zd\u0131r ve sabit bir ba\u015far\u0131<\/em> olas\u0131l\u0131\u011f\u0131na sahiptir.<\/p>\n \u00d6rne\u011fin, negatif binom da\u011f\u0131l\u0131m\u0131n\u0131 takip eden rastgele bir de\u011fi\u015fken, 6 say\u0131s\u0131 \u00fc\u00e7 kez at\u0131lana kadar bir zar\u0131n ka\u00e7 kez at\u0131lmas\u0131 gerekti\u011fidir. <\/p>\n![\"\\begin{array}{c}P[X=x_i]=p_i](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-115362012319df0fa040b1606f0cf461_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Ayr\u0131k olas\u0131l\u0131k da\u011f\u0131l\u0131mlar\u0131na \u00f6rnekler<\/span><\/h2>\n
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<\/span> Ayr\u0131k olas\u0131l\u0131k da\u011f\u0131l\u0131mlar\u0131n\u0131n t\u00fcrleri<\/span><\/h2>\n
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<\/span> Ayr\u0131k d\u00fczg\u00fcn da\u011f\u0131l\u0131m<\/span><\/h3>\n
![\"X\\sim](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-9977cf21c766a3d0ee2d79c8210dc598_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Bernoulli da\u011f\u0131l\u0131m\u0131<\/span><\/h3>\n
![\"\\begin{array}{c}X\\sim](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-384fd7d96d4d6584739b04a6e331b251_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Binom da\u011f\u0131l\u0131m\u0131<\/span><\/h3>\n
![\"X\\sim\\text{Bin}(n,p)\"](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-2f2b2be5bfe6c63bd13c552f4c893f59_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span>Bal\u0131k da\u011f\u0131t\u0131m\u0131<\/span><\/h3>\n
![\"X\\sim](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-be6e10a2b0137ec81fc7d366f237d1b2_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> \u00c7ok terimli da\u011f\u0131l\u0131m<\/span><\/h3>\n
<\/span>Geometrik da\u011f\u0131l\u0131m<\/span><\/h3>\n
![\"X\\sim\\text{Geom\\'etrica}(p)\"](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-22fef9b6ab8e3b351598caf9925c2b3f_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n<\/span> Negatif binom da\u011f\u0131l\u0131m\u0131<\/span><\/h3>\n
![\"X\\sim](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-171122de529a1c006bc46e8d89176016_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n
![\"X](\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-bd43d7c14739c66e63b224abf6cc20b3_l3.png\" "\\"Rendered")
<\/p>\n<\/p>\n