32 0 obj endobj 16 0 obj endobj >> 898 0 obj <>stream Let Sbe the set of real numbers that can be represented as repeating decimals of the form 0:abcwhere a;b;care distinct digits. (The Hat-Check Problem) h�bbd```b``>"w��0ɵ,^"-.��Ad,X�)D�l �6u`��`���$�րU��E�@�� �}Dʧ�ec��[��S��#���"����&&F�� �̠#�3�?� � �� � �ig`�`���`bk��1�"|2xf�BN`V �Df��oE��U�9��9��D͜mC4�Y�X�\@�1����|!�_~�{?��r�l������Ϡ���zzS�w@�N�[��A�lw�M���xX ������%��������.~��/�,(����7�.��c�A�p\Vw��?�/f���'����?�����������̯��t�Ӻ����U�>�s����YY�ŷ)�_�w���YΦ?���܎���ݜ2z��j�j�����ȼ����,�H����Ť�o�./����d��T�բW�{�� endobj Find the expected value of S. Problem 2.5 (AIME 2006 #6). Expected value Investment problem: • You have 100 dollars and can invest into a stock. 36 0 obj endobj Rather it is a weighted average of the possible values. The expected value is also called the mean or average of X and often denoted by µ (“mu”). ���Fn�5��I�O�+5h����Gl'y��u���X�����x��K+�hj?A���cr)����|�I F.�,�K�7E+�$?����� ��b�����@�XA�]��?������Ge�J���e��?�%��U��M0�ޫ�U�~X�ޖ���.�T���T�:/�I9�9�N�����N��,zÔ�� 3a�aW Probability, Expectation Value and Uncertainty We have seen that the physically observable properties of a quantum system are represented by Hermitean operators (also referred to as ‘observables’) such that the eigenvalues of the operator represents all the possible results that could be obtained if the associated physical observable were to be measured. ~�('`�"���P���zXLg�c�&Ƙ��+g݈\MO�~ԶZ*l�yfқZPiA���Fbh�~)d� s%~;�9f!���t�{)C�&��f訧5ܦmf�4#���I-�'[}���� �5�:��£\0����9��d!$/�pގ"F��e"�`����H��ჂI�3$i �j�Ժ�� %%EOF endobj Expected value is a summary statistic, providing a measure of the location or central tendency of a random variable. endobj 3. 12 0 obj (Equivalent Definitions of Expectation) If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 4. The returns are volatile and you may get either $120 with probability of 0.4, or $90 with probability 0.6. endobj << /S /GoTo /D (section.2) >> 871 0 obj <>/Filter/FlateDecode/ID[<591E06DC0FB1B742A9701B111D46F96F>]/Index[844 55]/Info 843 0 R/Length 124/Prev 238181/Root 845 0 R/Size 899/Type/XRef/W[1 3 1]>>stream The mean or expected value of X is defined by E(X) = sum x k p(x k). 44 0 obj << 1 6 = 7 2 This calculation shows that the name “expected value” is a little misleading; the random variable might never actually take on that value. << /S /GoTo /D [42 0 R /Fit ] >> endobj 5 0 obj 40 0 obj << /S /GoTo /D (subsection.4.2) >> h��YYS9���8��+b��a���`�Y�e(�v��qw�c~�d��P�ͼl��J�T��C�D�� �fRJ�5L���m��1),2DD ��@�BL���LE��b� 17 0 obj 37 0 obj (Expected Value of Two Dice) ,�Z3��=�}[�x�{��B�Ç��a}}?�eI�h(c,*��yz��]�ZG4>��Zm_hh|Z9�4��}�oi�ޠ��"�E4�!��v6�7�_cw�c�&�_C�Q ��k����B7+�JI�4mb�m��������[;Hw��@��H��U. • What is the expected value of your investment? All problems below were written by Lewis Chen. << /S /GoTo /D (subsection.4.1) >> Approach 1: Expected value The expected amount that you would earn from playing the game is 0.5( x)+0.5(10 x) This is bigger than 0 if 0.5( x)+0.5(10 x) 0 5 x Should pay at most $5 to play the game. 25 0 obj endobj 24 0 obj endobj /Length 2921 endobj endstream endobj startxref (Making a Baby Girl) Compute the expected value given a set of outcomes, probabilities, and payoffs If you're seeing this message, it means we're having trouble loading external resources on our website. endobj (The Chinese Appetizer Problem) endobj h�b```����Y� cc`a�����밿(��y��2��Z�w�Q@R� "��|�U�f�N���n�ʼ���-E�*̈��O�1������r�!�����i��]Y&z����,���'���P���=T�]�2 ����% j����}J@ %PDF-1.5 %���� (Mean Time to Failure) g!%���B��B��hnr���Ғ@�V������µ�j�y���M�e���8$�',�nx��/�C�� 2 ���p7^c�Th��@��� CC��O֊k�0���# endobj << /S /GoTo /D (subsection.2.2) >> endobj << /S /GoTo /D (subsection.2.1) >> x��ko����St��b��#)�Azw�I>�k٫ޮ䮴��r�������� �"G3�Ἱb��O���Ōѫ�n`�/�V����Y(��r���\|���+!�7f��1[�0j����2�嗢b���^������iv�خ�8q����O���06��z�6ݺyQV����]��Lk���ɹ}�˛�`{��LkȾy�e��"��M{��.���U��vi|�l��fK�zc��xؿU�D�������o&y8�|�W�Yf�q�.~����@r�]p�|���1g����B[ɸ����x}� �0�Y�ji ��rRif�Z�VJ�?�Y ǤG!5g��UB�|��8kH!�(�큘��w�%%�1e`c2�w�����^TUA�@ ���f�#P8�Sd�0��n[IY�;>EZ�t]�����5>D��sG������r�ķv�u|��&-u��r�. You can’t roll a 31 2 on an … 13 0 obj << /S /GoTo /D (section.4) >> 0 << /S /GoTo /D (section.3) >> endobj 2. 844 0 obj <> endobj << /S /GoTo /D (subsection.4.3) >> (Linearity of Expectation) 21 0 obj (Betting on Coins) 8 0 obj 9 0 obj /Filter /FlateDecode Find the sum of the elements of S. The next three problems are harder; in these problems linearity of expectation is not the main idea of the solution. << /S /GoTo /D (section.1) >> endobj 41 0 obj 28 0 obj stream endobj As seen in the above examples, the expected value need not be a possible value of the random variable. 33 0 obj %PDF-1.4 endstream endobj 845 0 obj <>/Metadata 49 0 R/Names 873 0 R/Outlines 89 0 R/PageMode/UseNone/Pages 842 0 R/StructTreeRoot 108 0 R/Type/Catalog>> endobj 846 0 obj <>/ProcSet[/PDF/Text]>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 847 0 obj <>stream Interpretations: (i) The expected value measures the center of the probability distribution - … 29 0 obj 20 0 obj endobj (An Expectation Paradox) A��i ���e�3݅;Ҁ��G��#��``��9���b`�R���աʚ w�ZM Expected Value and Variance Have you ever wondered whether it would be \worth it" to buy a lottery ticket every week, or pondered questions such as \If I were o ered a choice between a million dollars, or a 1 in 100 chance of a billion dollars, which would I choose?"

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