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					‫)ﻣﺤﻤﺪ اﻟﻜﯿﺎل(‬                         ‫اﻻﺣﺘﻤﺎﻻت‬

                                                                                        ‫‪Ë‬ﻣ ﻄﻠﺤﺎت‬
                   ‫ﻣﻌﻨﺎه‬                                          ‫اﻟﻤﺼﻄﻠﺢ اﻻﺣﺘﻤﺎﻟﻲ‬
          ‫ﻛﻞ ﺗﺠﺮﺑﺔ ﺗﻘﺒﻞ أﻛﺜﺮ ﻣﻦ ﻧﺘﯿﺠﺔ‬                              ‫ﺗﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ‬
 ‫ھﻲ ﻣﺠﻤﻮﻋﺔ اﻹﻣﻜﺎﻧﯿﺎت اﻟﻤﻤﻜﻨﺔ ﻟﺘﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ‬                       ‫‪ W‬ﻛﻮن اﻹﻣﻜﺎﻧﯿﺎت‬
          ‫‪ A‬ﺟﺰءا ﻣﻦ ﻛﻮن اﻹﻣﻜﺎﻧﯿﺎت ‪W‬‬                                    ‫ﺣﺪث ‪A‬‬
          ‫ﻛﻞ ﺣﺪث ﻳﺘﻀﻤﻦ ﻋﻨﺼﺮا وﺣﯿﺪا‬                                  ‫ﺣﺪث اﺑﺘﺪاﺋﻲ‬
      ‫إذا ﺗﺤﻘﻖ اﻟﺤﺪﺛﺎن ‪ A‬و ‪ B‬ﻓﻲ آن واﺣﺪ‬                          ‫ﺗﺤﻘﻖ اﻟﺤﺪث ‪A Ç B‬‬
          ‫إذا ﺗﺤﻘﻖ ‪ A‬أو ‪ B‬أو ھﻤﺎ ﻣﻌﺎ‬                             ‫ﺗﺤﻘﻖ اﻟﺤﺪث ‪A È B‬‬
   ‫) ‪A È A = W‬و ‪( A Ç A = Æ‬‬     ‫ھﻮ اﻟﺤﺪث ‪A‬‬                      ‫اﻟﺤﺪث اﻟﻤﻀﺎد ﻟﻠﺤﺪث ‪A‬‬
                ‫‪AÇB=Æ‬‬                                         ‫‪ A‬و ‪ B‬ﺣﺪﺛﺎن ﻏﯿﺮ ﻣﻨﺴﺠﻤﯿﻦ‬
                                                              ‫‪Ë‬اﺳﺘﻘﺮار ﺣﺪث - اﺣﺘﻤﺎل ﺣﺪث:‬
                                                                                       ‫ﺗﻌﺮﻳﻒ:‬
                                                     ‫ﻟﯿﻜﻦ ‪ W‬ﻛﻮن إﻣﻜﺎﻧﯿﺎت ﺗﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ‬
  ‫· ﻋﻨﺪﻣﺎ ﻳﺴﺘﻘﺮ اﺣﺘﻤﺎل ﺣﺪ ث اﺑﺘﺪاﺋﻲ }‪ {wi‬ﻓﻲ ﻗﯿﻤﺘﻪ ‪ pi‬ﻧﻘﻮل أن اﺣﺘﻤﺎل اﻟﺤﺪث }‪ {wi‬ھﻮ: ‪pi‬‬
                                                                                ‫وﻧﻜﺘﺐ: ‪P ({wi }) = pi‬‬
                       ‫· اﺣﺘﻤﺎل ﺣﺪث ھﻮ ﻣﺠﻤﻮع اﻻﺣﺘﻤﺎﻻت اﻻﺑﺘﺪاﺋﯿﺔ اﻟﺘﻲ ﺗﻜﻮن ھﺬا اﻟﺤﺪث‬
                  ‫أي إذا ﻛﺎن } ‪ A = {w1 ; w2 ; w3 ;...; wn‬ﺣﺪﺛﺎ ﻣﻦ ‪ W‬ﻓﺈن اﺣﺘﻤﺎل اﻟﺤﺪث ‪ A‬ھﻮ:‬
                    ‫) ‪p ( A ) = p ( w1 ) + p ( w2 ) + p ( w3 ) + ... + p ( wn‬‬
                                                                                     ‫ﺧﺎﺻﯿﺎت:‬
                                                             ‫ﻟﯿﻜﻦ ‪ W‬ﻛﻮن إﻣﻜﺎﻧﯿﺎت ﺗﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ‬
                                                                   ‫· 0 = ) ‪ p (Æ‬و 1 = )‪p ( W‬‬
                                                           ‫· 1 £ ) ‪ 0 £ p ( A‬ﻟﻜﻞ ﺣﺪث ‪ A‬ﻣﻦ ‪W‬‬
                                                                          ‫· اﺣﺘﻤﺎل اﺗﺤﺎد ﺣﺪﺛﯿﻦ:‬
                                                                         ‫ﻟﻜﻞ ﺣﺪﺛﯿﻦ ‪ A‬و ‪ B‬ﻣﻦ ‪W‬‬
                        ‫)‪p ( A È B) = p ( A ) + p ( B) - p ( A Ç B‬‬
               ‫) ‪ p ( A È B ) = p ( A ) + p ( B‬إذا ﻛﺎن ‪ A‬و ‪ B‬ﻏﯿﺮ ﻣﻨﺴﺠﻤﯿﻦ‬
                                                                         ‫· اﺣﺘﻤﺎل اﻟﺤﺪث اﻟﻤﻀﺎد:‬
                                                       ‫) (‬
                                                     ‫ﻟﻜﻞ ﺣﺪث ‪ A‬ﻣﻦ ‪ W‬ھﻮ: ) ‪p A = 1 - p ( A‬‬
                                                                  ‫‪Ë‬ﻓﺮﺿﯿﺔ ﺗﺴﺎوي اﻻﺣﺘﻤﺎﻻت:‬
                                                                                       ‫ﺗﻌﺮﻳﻒ:‬
        ‫إذا ﻛﺎﻧﺖ ﺟﻤﯿﻊ اﻷﺣﺪاث اﻻﺑﺘﺪاﺋﯿﺔ ﻣﺘﺴﺎوﻳﺔ اﻻﺣﺘﻤﺎل ﻓﻲ ﺗﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ ﻛﻮن إﻣﻜﺎﻧﯿﺘﮫﺎ ‪W‬‬
                                              ‫‪cardA‬‬
                                      ‫= )‪p(A‬‬          ‫ﻓﺈن اﺣﺘﻤﺎل ﻛﻞ ﺣﺪث ‪ A‬ﻣﻦ ‪ W‬ھﻮ:‬
                                              ‫‪cardW‬‬
                                             ‫‪Ë‬اﻻﺣﺘﻤﺎل اﻟﺸﺮﻃﻲ- اﺳﺘﻘﻼﻟﯿﺔ ﺣﺪﺛﯿﻦ:‬
                                                                                       ‫ﺗﻌﺮﻳﻒ:‬
                      ‫ﻟﯿﻜﻦ ‪ A‬و ‪ B‬ﺣﺪﺛﯿﻦ ﻣﺮﺗﺒﻄﯿﻦ ﺑﻨﻔﺲ اﻟﺘﺠﺮﺑﺔ اﻟﻌﺸﻮاﺋﯿﺔ ﺑﺤﯿﺚ: 0 ¹ ) ‪p ( A‬‬
                        ‫)‪p ( A Ç B‬‬
                   ‫) (‬
         ‫= ‪p ( B) = p B‬‬
                     ‫‪A‬‬     ‫)‪p(A‬‬
                                   ‫اﺣﺘﻤﺎل ﺣﺪث ‪ B‬ﻋﻠﻤﺎ أن اﻟﺤﺪث ‪ A‬ﻣﺤﻘﻖ ھﻮ اﻟﻌﺪد:‬
         ‫‪A‬‬
                                               ‫63‬
                                                                                   ‫ﻧﺘﯿﺠﺔ:‬
           ‫ﻟﻜﻞ ﺣﺪﺛﯿﻦ ‪ A‬و ‪ B‬ﻣﺮﺗﺒﻄﯿﻦ ﺑﻨﻔﺲ اﻟﺘﺠﺮﺑﺔ اﻟﻌﺸﻮاﺋﯿﺔ ﺑﺤﯿﺚ: 0 ¹ ) ‪p ( A ) ´ p ( B‬‬
                                  ‫‪p ( A Ç B) = p ( A ) ´ p B‬‬  ‫ﻟﺪﻳﻨﺎ: )‪( A ) = p ( B) ´ p ( A B‬‬
                                                                                   ‫ﺗﻌﺮﻳﻒ:‬
                                     ‫ﻟﻜﻞ ﺣﺪﺛﯿﻦ ‪ A‬و ‪ B‬ﻣﺮﺗﺒﻄﯿﻦ ﺑﻨﻔﺲ اﻟﺘﺠﺮﺑﺔ اﻟﻌﺸﻮاﺋﯿﺔ‬
                                  ‫) ‪ A Û p ( A Ç B ) = p ( A ) ´ p ( B‬و ‪ B‬ﺣﺪﺛﺎن ﻣﺴﺘﻘﻼن‬
                                                                                  ‫ﺧﺎﺻﯿﺔ:‬
                             ‫ﻟﯿﻜﻦ ‪ W‬ﻛﻮن إﻣﻜﺎﻧﯿﺎت ﺗﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ و 1‪ W‬و 2 ‪ W‬ﺗﺠﺰﻳﺌﺎ ل ‪W‬‬
                                                      ‫) ‪W1 È W2 = W‬و ‪( W1 Ç W2 = Æ‬‬
                                                                  ‫ﻟﻜﻞ ﺣﺪث ‪ A‬ﻣﻦ ‪: W‬‬
                    ‫‪p ( A ) = p ( W1 ) ´ p A‬‬‫) (‬
                                             ‫1‪W‬‬
                                                ‫‪+ p ( W2 ) ´ p A‬‬
                                                                 ‫2‪W‬‬‫) (‬
                                                              ‫‪Ë‬ﻗﺎﻧﻮن اﺣﺘﻤﺎل ﻣﺘﻐﯿﺮ ﻋﺸﻮاﺋﻲ:‬
                         ‫ﻟﯿﻜﻦ ‪ X‬ﻣﺘﻐﯿﺮا ﻋﺸﻮاﺋﯿﺎ ﻋﻠﻰ ‪ W‬ﻛﻮن إﻣﻜﺎﻧﯿﺎت ﺗﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ‬
                      ‫ﻟﺘﺤﺪﻳﺪ ﻗﺎﻧﻮن اﺣﺘﻤﺎل اﻟﻤﺘﻐﯿﺮ اﻟﻌﺸﻮاﺋﻲ ‪ X‬ﻧﺘﺒﻊ اﻟﻤﺮﺣﻠﺘﯿﻦ اﻟﺘﺎﻟﯿﺘﯿﻦ:‬
          ‫· ﺗﺤﺪﻳﺪ } ‪ : X ( W ) = {x1; x 2 ;x 3 ;...;x n‬ﻣﺠﻤﻮﻋﺔ اﻟﻘﯿﻢ اﻟﺘﻲ ﻳﺄﺧﺬھﺎ اﻟﻤﺘﻐﯿﺮ ‪X‬‬
                          ‫‪ i‬ﻣﻦ اﻟﻤﺠﻤﻮﻋﺔ }‪{1;2;...;n‬‬     ‫ﻧﺤﺴﺐ اﻻﺣﺘﻤﺎل ) ‪ p ( X = x i‬ﻟﻜﻞ‬        ‫·‬

                       ‫‪Ë‬اﻷﻣﻞ اﻟﺮﻳﺎﺿﻲ- اﻟﻤﻐﺎﻳﺮة- اﻻﻧﺤﺮاف اﻟﻄﺮازي ﻟﻤﺘﻐﯿﺮ ﻋﺸﻮاﺋﻲ:‬
                                                         ‫ﻟﯿﻜﻦ ‪ X‬ﻣﺘﻐﯿﺮا ﻋﺸﻮاﺋﯿﺎ ﻗﺎﻧﻮﻧﻪ‬
    ‫‪xi‬‬         ‫1‪x‬‬    ‫2‪x‬‬     ‫... 3 ‪x‬‬    ‫‪xn‬‬                    ‫ﻣﻌﺮف ﺑﺎﻟﺠﺪول اﻟﺘﺎﻟﻲ:‬
‫) ‪p(X = x i‬‬    ‫1‪p‬‬    ‫2‪p‬‬     ‫... 3‪p‬‬     ‫‪pn‬‬
                                                                                  ‫ﺗﻌﺎرﻳﻒ:‬
 ‫‪E ( X ) = x1 ´ p1 + x 2 ´ p 2 + x 3 ´ p3 + ... + x n ´ p n‬‬        ‫اﻷﻣﻞ اﻟﺮﻳﺎﺿﻲ ﻟﻠﻤﺘﻐﯿﺮ ‪X‬‬
              ‫² ‪V ( X ) = E ( X² ) - é E ( X ) ù‬‬
                                   ‫‪ë‬‬         ‫‪û‬‬
                                                                      ‫اﻟﻤﻐﺎﻳﺮة ﻟﻠﻤﺘﻐﯿﺮ ‪X‬‬

                    ‫)‪s(X) = V(X‬‬                                   ‫اﻻﻧﺤﺮاف اﻟﻄﺮازي ﻟﻠﻤﺘﻐﯿﺮ ‪X‬‬

                                                                             ‫‪Ë‬اﻟﻘﺎﻧﻮن اﻟﺤﺪاﻧﻲ:‬
                                                   ‫ﻟﯿﻜﻦ ‪ p‬اﺣﺘﻤﺎل ﺣﺪث ‪ A‬ﻓﻲ ﺗﺠﺮﺑﺔ ﻋﺸﻮاﺋﯿﺔ‬
                                                            ‫ﻧﻌﯿﺪ ھﺬه اﻟﺘﺠﺮﺑﺔ ‪ n‬ﻣﺮة‬
         ‫اﻟﻤﺘﻐﯿﺮ اﻟﻌﺸﻮاﺋﻲ ‪ X‬اﻟﺬي ﻳﺮﺑﻂ ﻛﻞ ﻧﺘﯿﺠﺔ ﺑﻌﺪد اﻟﻤﺮات اﻟﺘﻲ ﻳﺘﺤﻘﻖ ﻓﯿﮫﺎ اﻟﺤﺪث ‪A‬‬
                                                  ‫ﻳﺴﻤﻰ ﺗﻮزﻳﻌﺎ ﺣﺪاﻧﯿﺎ وﺳﯿﻄﺎه ‪ n‬و ‪p‬‬
                                                                                  ‫‪n -k‬‬
                          ‫) ‪"k Î {0;1;2;...;n} p ( X = k ) = Ck ´ p k ´ (1 - p‬‬
                                                            ‫‪n‬‬                            ‫وﻟﺪﻳﻨﺎ:‬
                                                   ‫‪E(X) = n ´ p‬‬                               ‫و‬
                                              ‫) ‪V ( X ) = np (1 - p‬‬                           ‫و‬




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