0000048072 00000 n @6f���P�d������Z�˥U}� 0000011795 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 0000076555 00000 n HHTTHT !3, THHTTT !2. Hence the two variables have covariance and correlation zero. 0000075183 00000 n > {���7ϱ�I��&���m�������'���}����G�O5��|J:��4�}�v$���:MRՌ �x��r=Z�iI�d���w+qTH}������~����,��~�w,5YZM�I4�C���)��ȣ`D��j\��Y�o�5��mM5�{)�T�[��u���ŵmm?A�հ=[\�mn\VW����iЇ�%�+��a�u64m��Z��Qz�q�����B���㦨�endstream 0000001136 00000 n Worked examples | Multiple Random Variables Example 1 Let X and Y be random variables that take on values from the set f¡1;0;1g. %PDF-1.3 No enrollment or registration. 5.1. Download files for later. » 0000065046 00000 n Note that although we sayX is 3.5 on the average, we must keep in mind that our X never actually equals 3.5 (in fact, it is impossible forX to equal 3.5). ]�ϼ�s��ܚi��Ւ���-��h�%%����l������~IJ�~ڄ�%��ckoh^�f'jA"��&����nf�n����~��݉��M�n�1:=�>��9' > Download from Internet Archive (MP4 - 24MB). Such a function, x, would be an example of a discrete random variable. Discrete Probability Distributions Let X be a discrete random variable, and suppose that the possible values that it can assume are given by x 1, x 2, x 3, . x��[I���� �v�×�m�hZ�/88XXa�c^��z�Ib���������7zz ���Z�����2���-���ѿ����67�-���� �� �=�|���6�u����Zq��|�Z��٣M���M�m�p�6۳g�/w�l��2�ww�jr�1�{���Z�^�j����z�')�v�o�lR� �|>7�#���݇s�����$�$��W���f���^p�i"ińQw�0�J*$������!Aw���Ϲ���-���l2�K�wOhT� p�0��8�{�Җ3v����ҿW�z � ��;���ǥOl)���4� Discrete Random Variables: Consider our coin toss again. 0000001694 00000 n �`I!�#��f%2��~\\v%���Z\�O1� Review the recitation problems in the PDF file below and try to solve them on your own. Probabilistic Systems Analysis and Applied Probability 0000002014 00000 n 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). Flash and JavaScript are required for this feature. Home 0000068740 00000 n > Download from Internet Archive (MP4 - 28MB), Joint Probability Mass Function (PMF) Drill 1, > Download from Internet Archive (MP4 - 57MB). 0000001824 00000 n stream The set of possible values of a random variables is known as itsRange. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. 0000077913 00000 n Use OCW to guide your own life-long learning, or to teach others. crete random variable while one which takes on a noncountably infinite number of values is called a nondiscrete random variable. Related to the probability mass function f X(x) = IP(X = x)isanotherimportantfunction called the cumulative distribution function (CDF), F X.Itisdefinedbytheformula Example 6-3: Consider the coin tossing experiment with S = {H, T}. 0000073670 00000 n Review the recitation problems in the PDF file below and try to solve them on your own. . 0000067011 00000 n The possible values of Xare 129, 130, and 131 mm. We don't offer credit or certification for using OCW. 0000064437 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. Discrete Random Variables This random variables can only take values between 0 and 6. » S�{��T���7�_���aLA ��0 Review the tutorial problems in the PDF file below and try to solve them on your own. 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. 0000058582 00000 n 0000003743 00000 n (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. 0000067188 00000 n 0000002074 00000 n With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. 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). 0000058398 00000 n 0 0000022155 00000 n 0000010064 00000 n 6 0 obj ��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@ 0000059918 00000 n 0000063790 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. �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. 29 0 obj xref EXAMPLE: Cars pass a roadside point, the gaps (in time) between successive cars being exponentially distributed. g��[�+Z�O�?��׏�d�p��>֬0Ƞ���9cR��@c�&�s�@�.>f1�v���:��qu����0N�E`�Jc,����� startxref �p}��@i$3C�Ґx�BJHf The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows: Each probability is between zero and one, inclusive (inclusive means to include zero and one). 0000023541 00000 n �h���K�J�g��K����ҋ��#�/'l�,mش'eO��V^:Y/i~3Y×V �(f&cdgayj��ШZՓ��h��jW=O+aFf��N]&_�m��ı�Yw����~/�R-�nT�e� �a@�4@g�$q������ `m�����q���ZOLY#�D�@ƃ��u����yX����8�m�V��\�E���e��J`��$��Q���[8�j���Ōʯו�,�a~�վz�������^�8�����fUe���u�"{���E~� m�XF�+�m`����Il��.��5OR�栛Q� 3225 trailer 0000075910 00000 n Then X(t;H) = sin(t), X(t;T) = cos(t) defines a discrete random … Courses The quantity (in the con-tinuous case – the discrete case is defined analogously) E(Xk) = Z∞ −∞ xkf(x)dx is called the kth moment of X. 5 0 obj stream Let Xdenote the length and Y denote the width. 0000062955 00000 n Unit I: Probability Models And Discrete Random Variables Send to friends and colleagues. :9 1�}~�����q�HY�zᅯ��8�rx�0D1��i�������^[즨��`ُ\��VNs&{k�K'z�ﱉ�6�+�-�\��6=�[�������g���a���'&m�Ho���p�� ��'{����6���"�';X��CΨ0��u�'9�>���"~X��b��3YE�XPx,����%��)$+�U�P�` I�$�tw������_�.�VP�c0�u��6P���'�E��|���@6�uvz;�����02H�/�Yم�`�퉵�"D�{����ȕRڔ3��p�? ™LJ�&. , arranged in some order. gX޺���Lف�b�aL��đS ���oi+��r5x�� ��RUĹ&�H�t���Fx]����Ӳ�}yU One of the problems has an accompanying video where a teaching assistant solves the same problem. 0000049395 00000 n ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. 0000073491 00000 n . Two of the problems have an accompanying video where a teaching assistant solves the same problem. endobj %PDF-1.2 %���� 0000047894 00000 n There's no signup, and no start or end dates. an example of a random variable. �v]��s�Yq\�/��Dh>���Id:�Q�J'QLy� �� �p��l�����v5u� 0000032340 00000 n �@}��i��� -�{@e�,�U��:[�5�2�10pM��'�3̶ �|p��&��e��"Q� G����i�K�. Made for sharing. ��Rz3��60�k�-�>$����. 0000002194 00000 n 0000018466 00000 n <> Learn more », © 2001–2018 147 42 We could have heads or tails as possible outcomes. Lecture 6: Discrete Random Variable Examples; Joint PMFs Video, > Download from Internet Archive (MP4 - 111MB). (b) Find a joint pmf assignment for which X and Y are not independent, but for which X2 and Y 2 are independent.