stream 6 0 obj 0000023729 00000 n » > Download from Internet Archive (MP4 - 28MB), Joint Probability Mass Function (PMF) Drill 1, > Download from Internet Archive (MP4 - 57MB). ��٧�|��$�#JDa�����˺����U"�)�'{��w۟�3�@��������E�#Y"`�Xh���S��b�c��hJX����b��U�*���u'?/��yF�~/�,i=�1�7�!a���7��9��8��iW����u�E�p�W���4#��e�|�����\�\*tVp7��=_�ژ}"?3��eV�3�y��w�G-�Z�ϧ��y�M6�/�"���m��#ᡈϗ�Gˢ��~dG/����U�h埾�;Hc�ۢ�o�2�AD@ The “moment generating function” gives us a nice way of collecting to-gether all the moments of a random varaible X into a single power series (i.e. 0000032340 00000 n ��4f5�A��W�"��x����*̄��&/�4V�^����\�~�>T�p�8"�hх�����u���ubv�Qϓ��Քz�F2�����ٟ�ܝ흇Q����t/��u����JU����6�u0.8Iy�a ’������_�qd�e��e��e Lecture 6: Discrete Random Variable Examples; Joint PMFs, Electrical Engineering and Computer Science, Probabilistic Systems Analysis and Applied Probability, Unit I: Probability Models And Discrete Random Variables, Unit IV: Laws Of Large Numbers And Inference, Lecture 6: Discrete Random Variable Examples; Joint PMFs Slides (PDF). x��ZI������!��z��Y�Հ/rG��D� 1b� ��F� ������&���,���=�l�X���"_tjН��߳g��ݣW;�}��^�t���?gϺ���;R�s�Ӈ�q��v��);�Н>�}�}���q���=�q��g����7GC�#�IEO�9���,kY����Ŕ�iJp.���<=LS|pA����?�QfÁ*"���o�)�4h`�n`yT��'�jv��˂�{8,�Upd9fBZ��Y��q�������,�qB99�W�Hu����{��N��N���W���,���/д�^���QR�%Q��`�����-Hd�. HHTTHT !3, THHTTT !2. 0000075183 00000 n Knowledge is your reward. Lecture 6. endobj For a discrete random process, probabilistic variable takes on only discrete values. Find materials for this course in the pages linked along the left. > {���7ϱ�I��&���m�������'���}����G�O5��|J:��4�}�v$���: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 0000001803 00000 n g��[�+Z�O�?��׏�d�p��>֬0Ƞ���9cR��@c�&�s�@�.>f1�v���:��qu����0N�E`�Jc,����� 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. ����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 an example of a random variable. 147 42 Two of the problems have an accompanying video where a teaching assistant solves the same problem. 0000003743 00000 n 0000010064 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. 0000076555 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). Note that although we sayX is 3.5 on the average, we must keep in mind that our X never actually equals 3.5 (in fact, it is impossible forX to equal 3.5). x�b```g``��������A���bl, L2@�f����x�����000*�t�{B��£�k�ˤE�3`s�46�Z�\ M���x��x���E���؏�$�%�N 4��8~D$Cqˢ3&��#C�=E�e�2Wu��̑P��&���nqYsUK���7���^���O� �)�dfR�����!�*6���a�$!/خ0f����AH S�{��T���7�_���aLA ��0 0000067188 00000 n MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. 0000075910 00000 n Flash and JavaScript are required for this feature. DISCRETE RANDOM VARIABLES 109 Remark5.3. 0000063790 00000 n . 0000018290 00000 n Let Xdenote the length and Y denote the width. Given a random variable X, let f(x) be its pdf. ]�ϼ�s��ܚi��Ւ���-��h�%%����l������~IJ�~ڄ�%��ckoh^�f'jA"��&����nf�n����~��݉��M�n�1:=�>��9' The set of possible values of a random variables is known as itsRange. 0000003924 00000 n �h���K�J�g��K����ҋ��#�/'l�,mش'eO��V^:Y/i~3Y×V �(f&cdgayj��ШZՓ��h��jW=O+aFf��N]&_�m��ı�Yw����~/�R-�nT�e� �a@�4@g�$q������ `m�����q���ZOLY#�D�@ƃ��u����yX����8�m�V��\�E���e��J`��$��Q���[8�j���Ōʯו�,�a~�վz�������^�8�����fUe���u�"{���E~� %%EOF endobj Types of random variable Most rvs are either discrete or continuous, but • one can devise some complicated counter-examples, and • there are practical examples of rvs which are partly discrete and partly continuous. Such a function, x, would be an example of a discrete random variable. But note that Xand Y are not inde- ... X(x)f Y(y) for all xand y. @6f���P�d������Z�˥U}� Courses , arranged in some order. . 0000011795 00000 n 0000058582 00000 n Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. 0000064437 00000 n Discrete Random Variables: Consider our coin toss again. 0000002074 00000 n EXAMPLE: Cars pass a roadside point, the gaps (in time) between successive cars being exponentially distributed. 0000049395 00000 n Discrete Random Variables x��[I���� �v�×�m�hZ�/88XXa�c^��z�Ib���������7zz ���Z�����2���-���ѿ����67�-���� �� �=�|���6�u����Zq��|�Z��٣M���M�m�p�6۳g�/w�l��2�ww�jr�1�{���Z�^�j����z�')�v�o�lR� �|>7�#���݇s�����$�$��W���f���^p�i"ińQw�0�J*$������!Aw���Ϲ���-���l2�K�wOhT� p�0��8�{�Җ3v����ҿW�z � ��;���ǥOl)���4� 0000033717 00000 n For every fixed value t = t0 of time, X(t0; ) is a discrete random variable. ��Rz3��60�k�-�>$����. 0000077913 00000 n Example 6-3: Consider the coin tossing experiment with S = {H, T}. 0000077221 00000 n The quantity (in the con-tinuous case – the discrete case is defined analogously) E(Xk) = Z∞ −∞ xkf(x)dx is called the kth moment of X. 0000001136 00000 n Massachusetts Institute of Technology. 0000032160 00000 n 5 0 obj �v]��s�Yq\�/��Dh>���Id:�Q�J'QLy� �� �p��l�����v5u� 0000011610 00000 n Modify, remix, and reuse (just remember to cite OCW as the source. 0000001694 00000 n Ou��_n�pi*���u�eL�u��B}V�ڝ_�&�]�-΋W[��}����� �m�9r�;`�$�5٢7�-2YB��P]�؉I/�b&�恒uI�PC��z,#L�`†�Б:��1�����v9�x5 ���̚�������f�a���v�p�w;�A-F k5��"�6h��v�d5-�3m�'.�D�j��p��a���Ԁ3��� ��_�^�n��Yu�$�r���X��>�X_�����8������nSIt?���}О��Ob�$ ��m*��C,�|m��ߧ� P .�G��vrAQÍ�~���NSJLi챐Enc�S��L�ª���탴3�.͟޿� ���Z��zR�F~T?.�%��( \�յx(��ŐT0���V^h����tLW�"E �i >�:�ap�}K��/B���Ih �:/Z�47���Ha���H��oqt^s'4e`�����:��cNH�X��v��e���e� ���؋ 0000062955 00000 n Learn more », © 2001–2018 29 0 obj ™LJ�&. 3225 startxref 0000073491 00000 n We could have heads or tails as possible outcomes. Review the Lecture 6: Discrete Random Variable Examples; Joint PMFs Slides (PDF) Read Sections 2.4–2.6 in the textbook; Recitation Problems and Recitation Help Videos. 0000065046 00000 n This is one of over 2,200 courses on OCW. Review the recitation problems in the PDF file below and try to solve them on your own. If we defined a variable, x, as the number of heads in a single toss, then x could possibly be 1 or 0, nothing else. 0000016865 00000 n 149 0 obj<>stream Related to the probability mass function f X(x) = IP(X = x)isanotherimportantfunction called the cumulative distribution function (CDF), F X.Itisdefinedbytheformula 0000073670 00000 n %"��(�r0I_JD�7�@�))h�)�ª� � » J��f��K���,���.��3��c��m��v>>I��[���E�A�thT�U�*�p~|86�j���u ���\� 0000058398 00000 n One of the problems has an accompanying video where a teaching assistant solves the same problem. No enrollment or registration. This random variables can only take values between 0 and 6. 147 0 obj<> endobj %�쏢 0 trailer Probabilistic Systems Analysis and Applied Probability "ϝ/�Vj�ə����V0m� �i&�b�h��"lXz����s��X��9��OJ�݃�?^cqR�Z旤#l��e�4��6o"7U� UFI'7�c 5Y�Y+ݍ=a�0���դ"P�M���������Eq 0000048072 00000 n 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). »

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