{"id":1323,"date":"2023-07-26T21:19:13","date_gmt":"2023-07-26T21:19:13","guid":{"rendered":"https:\/\/statorials.org\/tr\/pmf-istatistikleri\/"},"modified":"2023-07-26T21:19:13","modified_gmt":"2023-07-26T21:19:13","slug":"pmf-istatistikleri","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/pmf-istatistikleri\/","title":{"rendered":"I\u0307statistikte olas\u0131l\u0131k k\u00fctle fonksiyonu (pmf) nedir?"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\">Genellikle <strong>PMF<\/strong> olarak k\u0131salt\u0131lan bir <strong>olas\u0131l\u0131k k\u00fctle fonksiyonu<\/strong> , bize <a href=\"https:\/\/statorials.org\/tr\/rastgele-degiskenler\/\" target=\"_blank\" rel=\"noopener\">ayr\u0131 bir rastgele de\u011fi\u015fkenin<\/a> belirli bir de\u011feri alma olas\u0131l\u0131\u011f\u0131n\u0131 s\u00f6yler.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u00d6rne\u011fin bir zar\u0131 bir kez att\u0131\u011f\u0131m\u0131z\u0131 varsayal\u0131m. Zar\u0131n geldi\u011fi say\u0131y\u0131 x olarak kabul edersek, <em>x&#8217;in<\/em> farkl\u0131 de\u011ferlere e\u015fit olma olas\u0131l\u0131\u011f\u0131 \u015fu \u015fekilde a\u00e7\u0131klanabilir:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">P(X=1): 1\/6<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X=2): 1\/6<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X=3): 1\/6<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X=4): 1\/6<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X=5): 1\/6<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X=6): 1\/6<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Zarlar\u0131n 1 ile 6 aras\u0131nda herhangi bir say\u0131ya gelme \u015fans\u0131 e\u015fittir.<\/span><\/p>\n<p> <span style=\"color: #000000;\">Bu olas\u0131l\u0131klar\u0131 olas\u0131l\u0131k k\u00fctle fonksiyonu olarak \u015fu \u015fekilde yazabiliriz:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-12928 \" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/pmf1-1.png\" alt=\"Olas\u0131l\u0131k K\u00fctle Fonksiyonu \u00d6rne\u011fi\" width=\"227\" height=\"181\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Diyagram\u0131n sol taraf\u0131, sa\u011f taraftaki sonu\u00e7larla ili\u015fkili olas\u0131l\u0131\u011f\u0131 g\u00f6sterir:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-12929 \" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/pmf2.png\" alt=\"\u0130statistikte olas\u0131l\u0131k k\u00fctle fonksiyonu\" width=\"246\" height=\"232\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Olas\u0131l\u0131k k\u00fctle fonksiyonunun bir \u00f6zelli\u011fi, t\u00fcm olas\u0131l\u0131klar\u0131n toplam\u0131n\u0131n 1 olmas\u0131 gerekti\u011fidir. Bu PMF&#8217;nin bu ko\u015fulu kar\u015f\u0131lad\u0131\u011f\u0131n\u0131 fark edeceksiniz:<\/span><\/p>\n<p> <span style=\"color: #000000;\">Olas\u0131l\u0131klar\u0131n toplam\u0131 = 1\/6 + 1\/6 + 1\/6 + 1\/6 + 1\/6 + 1\/6 = 1.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><span style=\"color: #000000;\">Olas\u0131l\u0131k k\u00fctle fonksiyonu <strong>deste\u011fi,<\/strong> ayr\u0131k rastgele de\u011fi\u015fkenin alabilece\u011fi de\u011ferler k\u00fcmesini ifade eder. Bu \u00f6rnekte zar\u0131n de\u011feri bu de\u011ferlerden herhangi birini alabilece\u011fi i\u00e7in destek {1, 2, 3, 4, 5, 6} olacakt\u0131r.<\/span><\/span><\/p>\n<p> <span style=\"color: #000000;\"><span style=\"color: #000000;\"><span style=\"color: #000000;\">Deste\u011fin d\u0131\u015f\u0131nda PMF de\u011feri s\u0131f\u0131rd\u0131r. \u00d6rne\u011fin zar\u0131n \u201c0\u201d veya \u201c7\u201d veya \u201c8\u201d \u00fczerine gelme olas\u0131l\u0131\u011f\u0131 bu say\u0131lar\u0131n hi\u00e7biri parantez i\u00e7inde yer almad\u0131\u011f\u0131 i\u00e7in s\u0131f\u0131rd\u0131r.<\/span><\/span><\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Pratikte olas\u0131l\u0131k k\u00fctle fonksiyonlar\u0131<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Pratikte olas\u0131l\u0131k k\u00fctle fonksiyonlar\u0131n\u0131n en yayg\u0131n iki \u00f6rne\u011fi <a href=\"https:\/\/statorials.org\/tr\/binom-dagilimi-1\/\" target=\"_blank\" rel=\"noopener\">binom da\u011f\u0131l\u0131m\u0131<\/a> ve <a href=\"https:\/\/statorials.org\/tr\/balik-dagitimi\/\" target=\"_blank\" rel=\"noopener\">Poisson da\u011f\u0131l\u0131m\u0131 ile<\/a> ilgilidir.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>Binom da\u011f\u0131l\u0131m\u0131<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">E\u011fer bir <em>X<\/em> rastgele de\u011fi\u015fkeni binom da\u011f\u0131l\u0131m\u0131n\u0131 takip ediyorsa, <em>X<\/em> = <em>k<\/em> ba\u015far\u0131s\u0131n\u0131n olas\u0131l\u0131\u011f\u0131 a\u015fa\u011f\u0131daki form\u00fclle bulunabilir:<\/span><\/p>\n<p> <strong>P(X=k) = <sub>n<\/sub> C <sub>k<\/sub> * p <sup>k<\/sup> * (1-p) <sup>nk<\/sup><\/strong><\/p>\n<p> <span style=\"color: #000000;\">Alt\u0131n:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>n:<\/strong> deneme say\u0131s\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>k:<\/strong> ba\u015far\u0131 say\u0131s\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>p:<\/strong> belirli bir denemede ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong><sub>n<\/sub> C <sub>k<\/sub> :<\/strong> <em>n<\/em> denemede <em>k<\/em> ba\u015far\u0131 elde etmenin yollar\u0131n\u0131n say\u0131s\u0131<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">\u00d6rnek: Bir paray\u0131 3 kez att\u0131\u011f\u0131m\u0131z\u0131 varsayal\u0131m. Bu 3 at\u0131\u015fta 0, 1, 2 ve 3 tura gelme olas\u0131l\u0131\u011f\u0131n\u0131 belirlemek i\u00e7in yukar\u0131daki form\u00fcl\u00fc kullanabiliriz:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>P(X=0)<\/strong> = <sub>3<\/sub> C <sub>0<\/sub> * 0,5 <sup>0<\/sup> * (1-0,5) <sup>3-0<\/sup> = 1 * 1 * (0,5) <sup>3<\/sup> = <strong>0,125<\/strong><\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>P(X=1)<\/strong> = <sub>3<\/sub> C <sub>1<\/sub> * 0,5 <sup>1<\/sup> * (1-0,5) <sup>3-1<\/sup> = 1 * 1 * (0,5) <sup>2<\/sup> = <strong>0,375<\/strong><\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>P(X=2)<\/strong> = <sub>3<\/sub> C <sub>2<\/sub> * 0,5 <sup>2<\/sup> * (1-0,5) <sup>3-2<\/sup> = 1 * 1 * (0,5) <sup>1<\/sup> = <strong>0,375<\/strong><\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>P(X=3)<\/strong> = <sub>3<\/sub> C <sub>3<\/sub> * 0,5 <sup>3<\/sup> * (1-0,5) <sup>3-3<\/sup> = 1 * 1 * (0,5) <sup>0<\/sup> = <strong>0,125<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\"><strong>Bal\u0131k da\u011f\u0131t\u0131m\u0131<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Bir <em>X<\/em> rastgele de\u011fi\u015fkeni bir Poisson da\u011f\u0131l\u0131m\u0131n\u0131 takip ediyorsa, <em>X<\/em> = <em>k<\/em> ba\u015far\u0131s\u0131n\u0131n olas\u0131l\u0131\u011f\u0131 a\u015fa\u011f\u0131daki form\u00fclle bulunabilir:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=k) = \u03bb <sup>k<\/sup> * e <sup>\u2013 \u03bb<\/sup> \/ k!<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Alt\u0131n:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>\u03bb:<\/strong> belirli bir aral\u0131kta meydana gelen ortalama ba\u015far\u0131 say\u0131s\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>k:<\/strong> ba\u015far\u0131 say\u0131s\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><b>e:<\/b> yakla\u015f\u0131k 2,71828&#8217;e e\u015fit bir sabit<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">\u00d6rne\u011fin, belirli bir hastanede saatte ortalama 2 do\u011fum ger\u00e7ekle\u015fti\u011fini varsayal\u0131m. 0, 1, 2, 3 do\u011fum vb. ya\u015fanma olas\u0131l\u0131\u011f\u0131n\u0131 belirlemek i\u00e7in yukar\u0131daki form\u00fcl\u00fc kullanabiliriz. belirli bir saatte:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>P(X=0)<\/strong> = 2 <sup>0<\/sup> * e <sup>\u2013 2\/0<\/sup> ! = <strong>0,1353<\/strong><\/span><\/li>\n<li> <strong style=\"color: #000000;\">P(X=1)<\/strong> <span style=\"color: #000000;\">= 2<\/span> <sup style=\"color: #000000;\">1<\/sup> <span style=\"color: #000000;\">* e<\/span> <sup style=\"color: #000000;\">\u2013 2\/1<\/sup> <span style=\"color: #000000;\">! =<\/span> <strong style=\"color: #000000;\">0,2707<\/strong><\/li>\n<li> <strong style=\"color: #000000;\">P(X=2)<\/strong> <span style=\"color: #000000;\">= 2<\/span> <sup style=\"color: #000000;\">2<\/sup> <span style=\"color: #000000;\">* e<\/span> <sup style=\"color: #000000;\">\u2013 2\/2<\/sup> <span style=\"color: #000000;\">! =<\/span> <strong style=\"color: #000000;\">0,2707<\/strong><\/li>\n<li> <strong style=\"color: #000000;\">P(X=3)<\/strong> <span style=\"color: #000000;\">= 2<\/span> <sup style=\"color: #000000;\">3<\/sup> <span style=\"color: #000000;\">* e<\/span> <sup style=\"color: #000000;\">\u2013 2\/3<\/sup> <span style=\"color: #000000;\">! =<\/span> <strong style=\"color: #000000;\">0,1805<\/strong><\/li>\n<\/ul>\n<h3> <strong>PMF&#8217;yi g\u00f6r\u00fcnt\u00fcle<\/strong><\/h3>\n<p> <span style=\"color: #000000;\">Olas\u0131l\u0131k k\u00fctle fonksiyonlar\u0131n\u0131 s\u0131kl\u0131kla \u00e7ubuk grafiklerle g\u00f6rselle\u015ftiririz.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u00d6rne\u011fin, a\u015fa\u011f\u0131daki \u00e7ubuk grafik, \u00f6nceki \u00f6rnekte a\u00e7\u0131klanan Poisson da\u011f\u0131l\u0131m\u0131 i\u00e7in saat ba\u015f\u0131na do\u011fum say\u0131s\u0131yla ili\u015fkili olas\u0131l\u0131klar\u0131 g\u00f6stermektedir:<\/span> <\/p>\n<p><img decoding=\"async\" loading=\"lazy\" class=\"aligncenter wp-image-8131 \" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/poissondist1.png\" alt=\"Olas\u0131l\u0131k k\u00fctle fonksiyonu nas\u0131l g\u00f6rselle\u015ftirilir\" width=\"490\" height=\"360\" srcset=\"\" sizes=\"auto, \"><\/p>\n<p> <span style=\"color: #000000;\">Do\u011fum say\u0131s\u0131n\u0131n sonsuza kadar uzayabilece\u011fini, ancak 10&#8217;dan sonra olas\u0131l\u0131klar\u0131n o kadar azald\u0131\u011f\u0131n\u0131 ve bunlar\u0131 \u00e7ubuk grafikte bile g\u00f6remeyece\u011finizi unutmay\u0131n.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>PMF&#8217;nin \u00f6zellikleri<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Bir olas\u0131l\u0131k k\u00fctle fonksiyonu a\u015fa\u011f\u0131daki \u00f6zelliklere sahiptir:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>1. Destekte t\u00fcm olas\u0131l\u0131klar pozitiftir.<\/strong> \u00d6rne\u011fin bir zar\u0131n 1 ile 6 aras\u0131na d\u00fc\u015fme olas\u0131l\u0131\u011f\u0131 pozitif iken di\u011fer t\u00fcm sonu\u00e7lar\u0131n olas\u0131l\u0131\u011f\u0131 s\u0131f\u0131rd\u0131r.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>2. T\u00fcm sonu\u00e7lar\u0131n olas\u0131l\u0131\u011f\u0131 0 ile 1 aras\u0131ndad\u0131r.<\/strong> \u00d6rne\u011fin, zar\u0131n 1 ile 6 aras\u0131na d\u00fc\u015fme olas\u0131l\u0131\u011f\u0131 her sonu\u00e7 i\u00e7in 1\/6 veya 0,1666666&#8217;d\u0131r.<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>3. T\u00fcm olas\u0131l\u0131klar\u0131n toplam\u0131 1 olmal\u0131d\u0131r.<\/strong> \u00d6rne\u011fin bir zar\u0131n belirli bir say\u0131ya d\u00fc\u015fme olas\u0131l\u0131klar\u0131n\u0131n toplam\u0131 1\/6 + 1\/6 + 1\/6 + 1\/6 + 1\/6 + 1&#8217;dir. \/6 = 1.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Ek kaynaklar<\/strong><\/span><\/h3>\n<p> <a href=\"https:\/\/statorials.org\/tr\/rastgele-degiskenler\/\" target=\"_blank\" rel=\"noopener\">Rastgele de\u011fi\u015fkenler nelerdir?<\/a><br \/> CDF veya PDF: fark nedir?<br \/> <a href=\"https:\/\/statorials.org\/tr\/binom-dagilimi-1\/\" target=\"_blank\" rel=\"noopener\">Binom da\u011f\u0131l\u0131m\u0131na giri\u015f<\/a><br \/> <a href=\"https:\/\/statorials.org\/tr\/balik-dagitimi\/\" target=\"_blank\" rel=\"noopener\">Poisson da\u011f\u0131l\u0131m\u0131na giri\u015f<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Genellikle PMF olarak k\u0131salt\u0131lan bir olas\u0131l\u0131k k\u00fctle fonksiyonu , bize ayr\u0131 bir rastgele de\u011fi\u015fkenin belirli bir de\u011feri alma olas\u0131l\u0131\u011f\u0131n\u0131 s\u00f6yler. \u00d6rne\u011fin bir zar\u0131 bir kez att\u0131\u011f\u0131m\u0131z\u0131 varsayal\u0131m. Zar\u0131n geldi\u011fi say\u0131y\u0131 x olarak kabul edersek, x&#8217;in farkl\u0131 de\u011ferlere e\u015fit olma olas\u0131l\u0131\u011f\u0131 \u015fu \u015fekilde a\u00e7\u0131klanabilir: P(X=1): 1\/6 P(X=2): 1\/6 P(X=3): 1\/6 P(X=4): 1\/6 P(X=5): 1\/6 P(X=6): 1\/6 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-1323","post","type-post","status-publish","format-standard","hentry","category-rehber"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u0130statistikte olas\u0131l\u0131k k\u00fctle fonksiyonu (PMF) nedir?<\/title>\n<meta name=\"description\" content=\"Bu e\u011fitimde istatistikteki olas\u0131l\u0131k k\u00fctle fonksiyonuna (PMF) \u00f6rneklerle h\u0131zl\u0131 bir giri\u015f 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