0000067188 00000 n 0000011795 00000 n » trailer Probabilistic Systems Analysis and Applied Probability @6f���P�d������Z�˥U}� startxref > {���7ϱ�I��&���m�������'���}����G�O5��|J:��4�}�v$���:MRՌ �x��r=Z�iI�d���w+qTH}������~����,��~�w,5YZM�I4�C���)��ȣ`D��j\��Y�o�5��mM5�{)�T�[��u���ŵmm?A�հ=[\�mn\VW����iЇ�%�+��a�u64m��Z��Qz�q�����B���㦨�endstream Given a random variable X, let f(x) be its pdf. ™LJ�&. 0000011610 00000 n 0 Lecture 6: Discrete Random Variable Examples; Joint PMFs, Electrical Engineering and Computer Science, Probabilistic Systems Analysis and Applied Probability, Unit I: Probability Models And Discrete Random Variables, Unit IV: Laws Of Large Numbers And Inference, Lecture 6: Discrete Random Variable Examples; Joint PMFs Slides (PDF). 0000018466 00000 n Courses Discrete Random Variables 6 0 obj ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. ��4f5�A��W�"��x����*̄��&/�4V�^����\�~�>T�p�8"�hх�����u���ubv�Qϓ��Քz�F2�����ٟ�ܝ흇Q����t/��u����JU����6�u0.8Iy�a ’������_�qd�e��e��e 0000023729 00000 n Learn more », © 2001–2018 We don't offer credit or certification for using OCW. Use OCW to guide your own life-long learning, or to teach others. » Two of the problems have an accompanying video where a teaching assistant solves the same problem. x��[I���� �v�×�m�hZ�/88XXa�c^��z�Ib���������7zz ���Z�����2���-���ѿ����67�-���� �� �=�|���6�u����Zq��|�Z��٣M���M�m�p�6۳g�/w�l��2�ww�jr�1�{���Z�^�j����z�')�v�o�lR� �|>7�#���݇s�����$�$��W���f���^p�i"ińQw�0�J*$������!Aw���Ϲ���-���l2�K�wOhT� p�0��8�{�Җ3v����ҿW�z � ��;���ǥOl)���4� 0000067011 00000 n ]�ϼ�s��ܚi��Ւ���-��h�%%����l������~IJ�~ڄ�%��ckoh^�f'jA"��&����nf�n����~��݉��M�n�1:=�>��9' 0000000016 00000 n 0000075910 00000 n DISCRETE RANDOM VARIABLES 109 Remark5.3. 0000002074 00000 n 15.063 Summer 2003 1616 Continuous Random Variables A continuous random variable can take any value in some interval Example: X = time a customer spends waiting in line at the store • “Infinite” number of possible values for the random variable. 0000073491 00000 n �ŷMd��.P����d�v�r˿��ѹX�mR�LN@��>Վdep��XOd_��؄HN�¢�z�̅T �?���4�ħ���{���*�/�Ź��p�0Kr�P �2C�Y9 ��A�20�ݻ�����*���5'�����2ʖ37Ѽ(é�?�j*0fT���&m,�w��&�c��E �}y� ^v�y5"�U����F�X. 0000001803 00000 n The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows: Each probability is between zero and one, inclusive (inclusive means to include zero and one). 0000002194 00000 n 0000023541 00000 n Knowledge is your reward. 0000001694 00000 n S�{��T���7�_���aLA ��0 (b) Find a joint pmf assignment for which X and Y are not independent, but for which X2 and Y 2 are independent. Send to friends and colleagues. One of the problems has an accompanying video where a teaching assistant solves the same problem. 0000033717 00000 n endobj 0000065046 00000 n be described with a joint probability density function. <<83089bf982f7624f83e29cb71bce8b4d>]>> Lecture 6. No enrollment or registration. gX޺���Lف�b�aL��đS ���oi+��r5x�� ��RUĹ&�H�t���Fx]����Ӳ�}yU g��[�+Z�O�?��׏�d�p��>֬0Ƞ���9cR��@c�&�s�@�.>f1�v���:��qu����0N�E`�Jc,����� Modify, remix, and reuse (just remember to cite OCW as the source. �v]��s�Yq\�/��Dh>���Id:�Q�J'QLy� �� �p��l�����v5u� But note that Xand Y are not inde- ... X(x)f Y(y) for all xand y. Review the recitation problems in the PDF file below and try to solve them on your own. (a) Find a joint probability mass assignment for which X and Y are independent, and conflrm that X2 and Y 2 are then also independent. The quantity (in the con-tinuous case – the discrete case is defined analogously) E(Xk) = Z∞ −∞ xkf(x)dx is called the kth moment of X. 0000058398 00000 n 0000068740 00000 n �h���K�J�g��K����ҋ��#�/'l�,mش'eO��V^:Y/i~3Y×V �(f&cdgayj��ШZՓ��h��jW=O+aFf��N]&_�m��ı�Yw����~/�R-�nT�e� �a@�4@g�$q������ `m�����q���ZOLY#�D�@ƃ��u����yX����8�m�V��\�E���e��J`��$��Q���[8�j���Ōʯו�,�a~�վz�������^�8�����fUe���u�"{���E~� 147 0 obj<> endobj 0000003924 00000 n %%EOF This is one of over 2,200 courses on OCW. 0000016865 00000 n Home 0000059918 00000 n Electrical Engineering and Computer Science Hence the two variables have covariance and correlation zero. 3225 » %PDF-1.3 29 0 obj Massachusetts Institute of Technology. Example: Plastic covers for CDs (Discrete joint pmf) Measurements for the length and width of a rectangular plastic covers for CDs are rounded to the nearest mm(so they are discrete). Related to the probability mass function f X(x) = IP(X = x)isanotherimportantfunction called the cumulative distribution function (CDF), F X.Itisdefinedbytheformula Discrete Probability Distributions Let X be a discrete random variable, and suppose that the possible values that it can assume are given by x 1, x 2, x 3, . 0000049395 00000 n The set of possible values of a random variables is known as itsRange. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. :9 1�}~�����q�HY�zᅯ��8�rx�0D1��i�������^[즨��`ُ\��VNs&{k�K'z�ﱉ�6�+�-�\��6=�[�������g���a���'&m�Ho���p�� ��'{����6���"�';X��CΨ0��u�'9�>���"~X��b��3YE�XPx,����%��)$+�U�P�` I�$�tw������_�.�VP�c0�u��6P���'�E��|���@6�uvz;�����02H�/�Yم�`�퉵�"D�{����ȕRڔ3��p�? . » m�XF�+�m`����Il��.��5OR�栛Q� We could have heads or tails as possible outcomes. 0000032160 00000 n Example 6-3: Consider the coin tossing experiment with S = {H, T}. Download files for later. » The sum of the probabilities is one. > Download from Internet Archive (MP4 - 28MB), Joint Probability Mass Function (PMF) Drill 1, > Download from Internet Archive (MP4 - 57MB). Discrete Random Variables: Consider our coin toss again. 0000077221 00000 n 0000075183 00000 n crete random variable while one which takes on a noncountably infinite number of values is called a nondiscrete random variable. 0000002014 00000 n <> Types of random variable Most rvs are either discrete or continuous, but • one can devise some complicated counter-examples, and • there are practical examples of rvs which are partly discrete and partly continuous. This random variables can only take values between 0 and 6. ��٧�|��$�#JDa�����˺����U"�)�'{��w۟�3�@��������E�#Y"`�Xh���S��b�c��hJX����b��U�*���u'?/��yF�~/�,i=�1�7�!a���7��9��8��iW����u�E�p�W���4#��e�|�����\�\*tVp7��=_�ژ}"?3��eV�3�y��w�G-�Z�ϧ��y�M6�/�"���m��#ᡈϗ�Gˢ��~dG/����U�h埾�;Hc�ۢ�o�2�AD@ Review the tutorial problems in the PDF file below and try to solve them on your own. 0000022155 00000 n xref Flash and JavaScript are required for this feature. ����I3g(A��rnh��]P��6�!��4�^9�%��7F����� �M�PPE��mm!|˥����z��H��"&0J��)��1�Ѧ] v��-�D �)�6�(�������@�>��b��fb�q,�7Eq���{�&_Y�@D1#��z�ږ��*�P��@�|��������R�b���$R�Y���tݗ>��0n����g{��._Q�I5>Ei(���W\}�vZ>T��av�ᷠ�^;�k�u� ��j��(����!�_A&/��Lj���u�I�6W�Ψ�\�/�Nñ-c(�=�p��������#�6?�� q]���p�9�h]j���;yQ����=�e��5�X�E�)�v�t�Kd�����tgA��Z��=���A��]�]ܨ�oa��tF�׻ݨ^�aS�c��~;'�b���H��G�a�� ʹk:i��x��ƽnщ�����B�%��B� ��z֪�R�H�z+�����[DS� x�7c��r�@\]�O��P;�U1����|8n��T.���L�Ly�,��������H�!x{-}����M��� �cS��]���*�����czM�f�Td��)�K��n&��)I�����~y��*�����N

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