بعض التوزيات المستخدمة فى الاستدلال الالحصائى

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‫اﻟﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ‬

‫‪Normal Distribution‬‬

‫ﻳﻌﺘﺒــﺮ اﻟﺘﻮزﻳــﻊ اﻟﻄﺒﻴﻌــﻲ ﻣــﻦ أﻫــﻢ اﻟﺘﻮزﻳﻌــﺎت اﻻﺣﺘﻤﺎﻟﻴــﺔ اﻟﻤﺘﺼــﻠﺔ ﻓــﻲ ﻋﻠــﻢ اﻹﺣﺼــﺎء‪ ،‬ﺣﻴــﺚ‬

‫ﻳﺼــﻒ ﻛﺜﻴ ـﺮا ﻣــﻦ اﻟﻤﺠﺘﻤﻌــﺎت اﻟﻤﻮﺟــﻮدة ﻓــﻲ اﻟﻄﺒﻴﻌــﺔ ‪ ،‬اﻟﺼــﻨﺎﻋﺔ ‪ ،‬اﻷﺑﺤــﺎث‪ .‬داﻟــﺔ ﻛﺜﺎﻓــﺔ اﻻﺣﺘﻤــﺎل‬ ‫ﻟﻠﺘﻮزﻳﻊ اﻟﻄﺒﻴﻌﻲ ﻫﻲ‪:‬‬ ‫‪( x  ) 2‬‬ ‫‪2 2‬‬

‫‪‬‬

‫‪1‬‬ ‫‪e‬‬ ‫‪ 2‬‬

‫‪f (x) ‬‬

‫ﻋﻠ ـ ـﻰ اﻟﻔﺘـ ــﺮة ‪    x  ‬ﺣﻴـ ــﺚ …‪ e=2.71828‬و …‪  =3.14159‬و ‪ ‬و ‪ ‬ﻫﻤـ ــﺎ‬

‫اﻟﻮﺳـﻂ اﻟﺤﺴــﺎﺑﻲ ) اﻟﻤﺘﻮﺳـﻂ ( واﻻﻧﺤـﺮاف اﻟﻤﻌﻴــﺎري ﻋﻠـﻰ اﻟﺘــﻮاﻟﻲ‪ .‬ﺑﻴــﺎن )‪ f(x‬ﻓـﻲ اﻟﺸــﻜﻞ اﻟﺘــﺎﻟﻰ‬ ‫ﻋﻠﻰ ﺷـﻜﻞ ﻧـﺎﻗﻮس ﻣﺘﻤﺎﺛـﻞ ﺣـﻮل اﻟﻌﻤـﻮد اﻟﻤﻘـﺎم ﻋﻠـﻰ اﻟﻮﺳـﻂ اﻟﺤﺴـﺎﺑﻰ‪ ٠‬ﻳﻨﻄﺒـﻖ اﻟﻮﺳـﻂ اﻟﺤﺴـﺎﺑﻲ‬

‫ﻋﻠـﻰ اﻟﻮﺳـﻴﻂ وأﻳﻀـﺎ ﻋﻠـﻰ اﻟﻤﻨـﻮال‪ ٠‬ﻳﺘﻘـﺎرب ﻃﺮﻓـﺎ ﻣﻨﺤﻨـﻰ اﻟﺘﻮزﻳـﻊ اﻟﻄﺒﻴﻌـﻲ ﻣـﻦ اﻟﺼـﻔﺮ ﻋﻨـﺪﻣﺎ‬ ‫‪ x  ‬أو ‪٠ x  ‬‬

‫ﻷي ﺗﻮزﻳﻊ ﻃﺒﻴﻌﻲ ﻓﺈن ‪-:‬‬

‫)ا( اﻟﻤﺴﺎﺣﺔ اﻟﻮاﻗﻌﺔ ﺗﺤﺖ اﻟﻤﻨﺤﻨﻲ اﻟﻄﺒﻴﻌﻲ ﺑﻴﻦ اﻟﻨﻘﻄﺘﻴﻦ ‪ ‬و )‪ (  + ‬ﺗﻤﺜﻞ ‪34.13%‬‬

‫ﻣﻦ اﻟﻤﺴﺎﺣﺔ اﻟﻜﻠﻴﺔ ﻛﻤﺎ ﻳﺘﻀﺢ ﻣﻦ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻰ‪.‬‬

‫)ب( اﻟﻤﺴﺎﺣﺔ اﻟﻮاﻗﻌﺔ ﺗﺤﺖ اﻟﻤﻨﺤﻨﻰ اﻟﻄﺒﻴﻌـﻲ ﺑـﻴﻦ اﻟﻨﻘﻄﺘـﻴﻦ ‪ ‬و )‪ ( +2‬ﺗﻤﺜـﻞ ‪ 47.72‬ﻣـﻦ‬

‫اﻟﻤﺴﺎﺣﺔ اﻟﻜﻠﻴﺔ ﻛﻤﺎ ﻳﺘﻀﺢ ﻣﻦ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻰ‪.‬‬

‫)ﺟـ( اﻟﻤﺴﺎﺣﺔ اﻟﻮاﻗﻌﺔ ﺗﺤﺖ اﻟﻤﻨﺤﻨﻰ اﻟﻄﺒﻴﻌﻲ ﺑﻴﻦ اﻟﻨﻘﻄﺘﻴﻦ ‪ ‬و )‪ ( +3‬ﺗﻤﺜـﻞ ‪ 49.86 %‬ﻣـﻦ‬

‫اﻟﻤﺴﺎﺣﺔ اﻟﻜﻠﻴﺔ ﻛﻤﺎ ﻳﺘﻀﺢ ﻣﻦ اﻟﺸﻜﻞ اﻟﺘﺎﻟﻰ‪.‬‬

‫‪١‬‬


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