Your Federal Quarterly Tax Payments are due April 15th

probabite by AbdrrahimEddafi

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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

‫73‬

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