endobj 35 0 obj Tutorial 3: Probability Theory, Random Variables and Distributions 3 Task 4: Note the following: If event A is partitioned by a series of n subsets B i then P(A) = P i P(A\B i). 750 708.3 722.2 763.9 680.6 652.8 784.7 750 361.1 513.9 777.8 625 916.7 750 777.8 >> }j�ԩS�:�C�B@n���r绰-& دm����CUn}�����*4]��C'�]�>7�&--���p���(�Q���[��bvߗ!��ćT2�V��}6 ��äU �0�{�u�8Z�;fr���w6�U��w�vw�(���#@�K(�� ��T��l���i�gJ10P��3� p1W\,V��Ā� ����*�5�wʎ��?K穃��{^�!�6'ے��t��]�6� ��fc�+�B�2��(�I9���djlX3��{#*��ϢI�\$-7������l�u����58=��q}_q%�H���f�\�9^���|����a� W!��f��� ���8X9v {! 511.1 575 1150 575 575 575 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Name/F4 20 0 obj /Name/F3 /FirstChar 33 277.8 500] Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 stream 863.9 786.1 863.9 862.5 638.9 800 884.7 869.4 1188.9 869.4 869.4 702.8 319.4 602.8 /Type/Font 766.7 715.6 766.7 0 0 715.6 613.3 562.2 587.8 881.7 894.4 306.7 332.2 511.1 511.1 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 525 386.1 885.5 591.1 591.1 885.5 865.5 816.7 826.7 875.5 756.7 727.2 895.3 896.1 471.7 /BaseFont/ASMHGU+CMTI10 << 436.1 552.8 844.4 319.4 377.8 319.4 552.8 552.8 552.8 552.8 552.8 552.8 552.8 552.8 Probability Tutorial. stream /BaseFont/RHERLQ+CMBX10 endstream << 26 0 obj /Type/Font Note: Some other pre-requisites (e.g., concepts from information theory, linear algebra, optimization, etc.) 6 0 obj 277.8 305.6 500 500 500 500 500 808.6 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 << endobj /Widths[277.8 500 833.3 500 833.3 777.8 277.8 388.9 388.9 500 777.8 277.8 333.3 277.8 << /Filter[/FlateDecode] /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/sterling/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi << x�}��n�0��HH�M� �̌�& J2@ߞk; ")����|�ܟ�8'_I:ސ�����\I�G�IAv R�?��b'��Ua�^Ʈ�9��k�臾;ԇpc�>�������K�����I���0�HC�F���L�SA�`N�\$] ����IAG��aFG,��*V�I#�����4�R��}h��됹f���S�c�Ȃ�L�������p�I��3hӆ1tmN��dB> V,�L�t%7��_�k���5Dcg]�R�m:�iJ��NZG��d '��'�!~x�����m�r�s�LW ��=��HA>a�s�e������P���Ad�2�k�c����=�&�|,��5����8f @�(-�*��մqI�+�.C����Z�X��K�l���/�1�(J��94��>.y�0�2����� /LastChar 196 865.5 708.9 356.1 620.6 356.1 591.1 355.6 355.6 591.1 532.2 532.2 591.1 532.2 400 2013 Probability Theory Tutorial 1. 30 0 obj 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 Probability theory is important to empirical sci-entists because it gives them a rational frame w ork to mak e inferences and test hypotheses based on uncertain empirical data. University. /FontDescriptor 41 0 R /Name/F1 abilityÓ, ÒchanceÓ and ÒriskÓ. endobj 7 0 obj 319.4 575 319.4 319.4 559 638.9 511.1 638.9 527.1 351.4 575 638.9 319.4 351.4 606.9 /Name/F2 575 1041.7 1169.4 894.4 319.4 575] /Widths[622.5 466.3 591.4 828.1 517 362.8 654.2 1000 1000 1000 1000 277.8 277.8 500 Let A be the set of hearts, and B be the set of even cards (counting 2, 4, 6, 8, 10 and Queen as even cards). 750 758.5 714.7 827.9 738.2 643.1 786.2 831.3 439.6 554.5 849.3 680.6 970.1 803.5 36 0 obj /FirstChar 33 /Encoding 7 0 R 824.4 635.6 975 1091.7 844.4 319.4 319.4 552.8 902.8 552.8 902.8 844.4 319.4 436.1 endobj >> 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 >> /Type/Font 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 13 0 obj >> /Subtype/Type1 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 Basic probability theory • Definition: Real-valued random variableX is a real-valued and measurable function defined on the sample space Ω, X: Ω→ ℜ – Each sample point ω ∈ Ω is associated with a real number X(ω) • Measurabilitymeans that all sets of type belong to the set of events , that is {X ≤ x} ∈ Economics (101) Academic year. Tutorial on probability theory A.1 Random events Probability theory is fundamental to much of our discussion in this book. << /FirstChar 33 743.3 743.3 613.3 306.7 514.4 306.7 511.1 306.7 306.7 511.1 460 460 511.1 460 306.7 /F2 13 0 R What is P(B)? 532.2 767.8 560.6 561.7 490.6 591.1 1182.2 591.1 591.1 591.1 0 0 0 0 0 0 0 0 0 0 /F4 20 0 R >> 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 500 500 500 500 500 500 500 500 500 500 500 277.8 277.8 777.8 500 777.8 500 530.9 4. 525 525] 306.7 766.7 511.1 511.1 766.7 743.3 703.9 715.6 755 678.3 652.8 773.6 743.3 385.6 << /Type/Encoding Basic Concepts Random Experiment. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 697.8 944.4 885.5 806.7 767.8 896.1 /FirstChar 33 /Encoding 7 0 R 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] Discrete random variable is a variable that can be any value in a set (sample space), which has a finite number of values. >> University of the Witwatersrand, Johannesburg. 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] 2013/2014. endobj endobj /Subtype/Type1 306.7 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 511.1 306.7 306.7 /Subtype/Type1 Sign in Register; Hide. endobj 277.8 305.6 500 500 500 500 500 750 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 638.9 638.9 958.3 958.3 319.4 351.4 575 575 575 575 575 869.4 511.1 597.2 830.6 894.4 /F3 16 0 R /F5 23 0 R /Name/F5 /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 500 555.6 527.8 391.7 394.4 388.9 555.6 527.8 722.2 527.8 527.8 444.4 500 1000 500 Probability theory is also useful to engineers building systems that ha … /Type/Font 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 525 525 525 525 525 525 525 525 525 525 0 0 525 /FontDescriptor 12 0 R /Subtype/Type1 591.1 591.1 591.1 591.1 948.9 532.2 665 826.7 826.7 591.1 1022.8 1140.5 885.5 296.7 >> 255/dieresis] /FirstChar 33 388.9 1000 1000 416.7 528.6 429.2 432.8 520.5 465.6 489.6 477 576.2 344.5 411.8 520.6 /Differences[33/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi