{"id":1384,"date":"2018-04-30T19:36:18","date_gmt":"2018-04-30T16:36:18","guid":{"rendered":"http:\/\/www.akademikidea.org\/a-kitap\/?p=1384"},"modified":"2025-08-08T22:51:05","modified_gmt":"2025-08-08T19:51:05","slug":"normal-olasilik-yogunluk-islevi","status":"publish","type":"post","link":"https:\/\/www.akademikidea.org\/a-kitap\/normal-olasilik-yogunluk-islevi\/","title":{"rendered":"Normal Olas\u0131l\u0131k Yo\u011funluk \u0130\u015flevi"},"content":{"rendered":"

\u015eekil 1.’de g\u00f6sterildi\u011fi gibi yatay bir d\u00fczlem \u00fczerinde dikd\u00f6rtgensel bir koordinat dizgesinin ba\u015flang\u0131\u00e7 noktas\u0131na ni\u015fan al\u0131narak yap\u0131lan \u00e7ok say\u0131daki at\u0131\u015ftan bir tanesinin \u00a0x<\/em> ekseniyle bir \\(\\theta\\) a\u00e7\u0131s\u0131 do\u011frultusunda ve\u00a0ba\u015flang\u0131\u00e7 noktas\u0131ndan\u00a0r<\/em>\u00a0 birim uzakl\u0131ktaki bir noktaya isabet etti\u011fini d\u00fc\u015f\u00fcnelim. Ayr\u0131ca, hedeften b\u00fcy\u00fck sapmalar\u0131n k\u00fc\u00e7\u00fck sapmalara g\u00f6re daha az olas\u0131 oldu\u011funu ve dik y\u00f6nlerdeki sapmalar\u0131n bir birinden ba\u011f\u0131ms\u0131z oldu\u011funu varsayal\u0131m.<\/p>\n

\n
\n
\n
\n

\"\"<\/a>Hedef tahtas\u0131na isabet eden noktan\u0131n \\(x\\)’ten \\(x +\\Delta x \\)’e kadar olan dilim i\u00e7inde olma olas\u0131l\u0131\u011f\u0131 \\( o(x) \\Delta x \\) ve \\(y\\)’den \\(y +\\Delta y \\)’e kadar olan dilim i\u00e7inde olma olas\u0131l\u0131\u011f\u0131 \\( o(y) \\Delta y\\) olsun. Hedefin merkezinden \\(r \\) kadar sapan isabet noktalar\u0131 koordinatlardan ba\u011f\u0131ms\u0131z oldu\u011fundan, isabet noktas\u0131n\u0131n \\(\\Delta x \\) ve \\( \\Delta y \\) geni\u015flikli dilimlerin kesi\u015fimi olan dikd\u00f6rtgen alana d\u00fc\u015fme olas\u0131l\u0131\u011f\u0131<\/p>\n

$$ o(x)o(y) \\Delta x \\Delta y=g(r)\\Delta x \\Delta y $$<\/p>\n

olur. Buradan<\/p>\n

$$ g(r)=o(x)o(y) $$<\/p>\n

oldu\u011fu \u00e7\u0131kar.<\/p>\n

\\( g(r) \\) kutupsal koordinatlarda \\(\\theta \\) a\u00e7\u0131s\u0131na ba\u011fl\u0131 olmad\u0131\u011f\u0131ndan k\u0131smi t\u00fcrevler kullan\u0131larak<\/p>\n

$$ \\frac{\\partial g(r)}{\\partial \\theta}=0=o(x) \\frac{\\partial o(y)}{\\partial \\theta}+ o(y)\\frac{\\partial o(x)}{\\partial \\theta}$$<\/p>\n

yaz\u0131labilir. Kutupsal<\/p>\n

$$x=r \\cos \\theta$$<\/p>\n

$$y=r \\sin \\theta$$<\/p>\n

ili\u015fkilerine g\u00f6re,<\/p>\n

$$\\frac {\\partial o(x)}{\\partial \\theta}=\\frac {\\partial o(x)}{\\partial x}\\frac {\\partial x}{\\partial \\theta}=o{}'(x)(-y)$$<\/p>\n

$$\\frac {\\partial o(y)}{\\partial \\theta}=\\frac {\\partial o(y)}{\\partial y}\\frac {\\partial y}{\\partial \\theta}=o{}'(y)(x)$$<\/p>\n

ve buradan da,<\/p>\n

$$o(x){o}'(y)(x)-o(y){o}'(x)(y)=0$$<\/p>\n

olur. De\u011fi\u015fgenler ayr\u0131larak,\u00a0\\(x\\) ve \\(y\\) de\u011fi\u015fgenleri ba\u011f\u0131ms\u0131z oldu\u011fundan her iki yan\u0131 \\(K\\) gibi bir sabite e\u015fit olmak zorunda olan,<\/p>\n

$$\\frac{{o}'(x)}{xo(x)}=\\frac{{o}'(y)}{yo(y)}=K$$<\/p>\n

t\u00fcrevsel denklemleri elde edilir. Dolay\u0131s\u0131 ile, \\(x\\) de\u011fi\u015fgeni i\u00e7in,<\/p>\n

$$\\frac{{o}'(x)}{o(x)}=Kx$$<\/p>\n

$$\\ln o(x)=K \u00a0x^2\/2 + C$$<\/p>\n

$$o(x)=Ae^{K x^2\/2}$$<\/p>\n

olur. Ancak k\u00fc\u00e7\u00fck sapmalar\u0131n b\u00fcy\u00fck olanlardan daha az olas\u0131 oldu\u011fu varsay\u0131ld\u0131\u011f\u0131ndan \\(K<0\\) olur ve \\(K=-1\\)\u00a0olarak al\u0131n\u0131rsa,<\/p>\n

$$o(x)=Ae^{-x^2 \/ 2 }…………………………………………(1)$$<\/p>\n

olur. \u00a0Olas\u0131l\u0131k i\u015flevi tan\u0131m\u0131na g\u00f6re,<\/p>\n

$$ \\int_{-\\infty }^{\\infty }Ae^{-x^2 \/ 2 }dx=1$$<\/p>\n

ya da,<\/p>\n

$$ A=\\frac {1}{\\int_{-\\infty }^{\\infty }e^{-x^2 \/ 2 }}dx $$<\/p>\n

olmal\u0131d\u0131r. \\(A\\)’n\u0131n tan\u0131m\u0131ndaki t\u00fcmlev, t\u00fcmlevin karesi al\u0131narak elde edilen,<\/p>\n

$$T^2= \\int_{-\\infty }^{\\infty }e^{-x^2 \/ 2 }dx \\int_{-\\infty }^{\\infty }e^{-y^2 \/ 2 }dy=\\int_{-\\infty}^{\\infty}\\int_{-\\infty }^{\\infty }e^{-(x^2+y^2) \/ 2 }dxdy$$<\/p>\n

bi\u00e7imindeki \u00e7ift t\u00fcmlev de\u011feri hesaplanabilir. Kutupsal koordinat sistemine ge\u00e7ilerek hesaplanan,<\/p>\n

$$T^2=\\int_{0}^{2\\pi}\\int_{0 }^{\\infty }e^{-r^2 \/ 2 }rdrd \\theta=2 \\pi\u00a0\\int_{0 }^{\\infty }e^{-r^2 \/ 2 \u00a0}rdr=2 \\pi (-e^{-r^2 \/ 2 \u00a0}) |_{0}^{\\infty}=2\\pi$$<\/p>\n

de\u011ferinden,<\/p>\n

$$A=1\/\\sqrt {2 \\pi}$$<\/p>\n

olarak bulunur. Denklem (1)’de \\(A\\)’n\u0131n de\u011feri yerine konursa, birim normal da\u011f\u0131l\u0131m olas\u0131l\u0131k yo\u011funluk i\u015flevi<\/p>\n

$$o(x)=\\frac {1}{\\sqrt {2 \\pi}}e^{-x^2 \/ 2 }………………………………….(2)$$<\/p>\n

bi\u00e7iminde elde edilmi\u015f olur.<\/p>\n

John F. W. Herschel (1792-1871)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n

\\(\\left\\{ {{x_n} = {x_{n – 1}} + {x_{n – 2}},{x_0} = 0,{x_1} = 1,n = 0,1,2, \\cdots } \\right\\}\\)<\/p>\n

\\(\\frac{n!}{r!(n-r)!} \\sum_{i=1}^{n}{(X_i – \\overline{X})^2}\\)<\/p>\n<\/div>\n

Kaynaklar<\/em><\/h3>\n
\n
\n
\n
\n
\n
\n

Pearson, Karl\u00a0(1924)<\/a> Note on the Origin of the Normal Curve of Errors, Biometrika<\/em>, 16(3\/4):402-404.<\/p>\n

Grossman, Stanley I. (1986)<\/a> \u00a0Multivariable calculus, linear algebra, and differential equations,<\/em> Academic Press, \u00a0Elsevier Inc. : 297-298.<\/p>\n

Hamming, Richard W. (1991)<\/a> The Art of Probability_ For Scientists and Engineers , Addison-Wesley.:208-211.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Normal distribution’s probability density function derived in 5 minutes<\/a><\/p>\n

Derivation of the Normal (Gaussian) Distribution<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

\u015eekil 1.’de g\u00f6sterildi\u011fi gibi yatay bir d\u00fczlem \u00fczerinde dikd\u00f6rtgensel bir koordinat dizgesinin ba\u015flang\u0131\u00e7 noktas\u0131na ni\u015fan al\u0131narak yap\u0131lan \u00e7ok say\u0131daki at\u0131\u015ftan bir tanesinin \u00a0x ekseniyle bir \\(\\theta\\) a\u00e7\u0131s\u0131 do\u011frultusunda ve\u00a0ba\u015flang\u0131\u00e7 noktas\u0131ndan\u00a0r\u00a0 birim uzakl\u0131ktaki bir noktaya isabet etti\u011fini d\u00fc\u015f\u00fcnelim. Ayr\u0131ca, hedeften b\u00fcy\u00fck sapmalar\u0131n k\u00fc\u00e7\u00fck sapmalara g\u00f6re daha az olas\u0131 oldu\u011funu ve dik y\u00f6nlerdeki sapmalar\u0131n bir birinden ba\u011f\u0131ms\u0131z…<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[],"class_list":["post-1384","post","type-post","status-publish","format-standard","hentry","category-istatistik"],"_links":{"self":[{"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/posts\/1384","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/comments?post=1384"}],"version-history":[{"count":70,"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/posts\/1384\/revisions"}],"predecessor-version":[{"id":2423,"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/posts\/1384\/revisions\/2423"}],"wp:attachment":[{"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/media?parent=1384"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/categories?post=1384"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.akademikidea.org\/a-kitap\/wp-json\/wp\/v2\/tags?post=1384"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}