6.041 / 6.431 1. Probability Models and Axioms

L01p.1 L01p.2 6.041 Probabilistic Systems Analysis Coursework 6.431 Applied Probability – Quiz 1 (October 12, 7:30-9:00pm) 20% Staff• – Quiz 2 (November 2, 7:30-9:30pm) 28% – Lecturer: John Tsitsiklis, – Final exam (scheduled by registrar) 38% – Recitation instructors: Dimitri Bertsekas (6.431), Peter Hagelstein, Ali Shoeb, Vivek Goyal – Weekly homework (best 9 of 10) 9% – Head TA: Shashank Dwivedi, – Attendance/participation/enthusiasm in 5% – Other TAs: Alia Atwi, Uzoma Orji, Sam Zamanian recitations/tutorials Pick up and read course information handout • • Turn in recitation and tutorial scheduling form • Collaboration policy described in course info handout (last sheet of course information handout)Text: Introduction to Probability, 2nd Edition,Pick up copy of slides • • D. P. Bertsekas and J. N. Tsitsiklis, Athena Scientific, 2008 Read the text! L01 p. 3 LECTURE 1 • Readings: Sections 1.1, 1.2 Lecture outline • Probability as a mathematical framework for reasoning about uncertainty • Probabilistic models – sample space – probability law • Axioms of probability • Simple examples L01 p. 4 Sample space " • “List” (set) of possible outcomes • List must be: – Mutually exclusive – Collectively exhaustive • Art: to be at the “right” granularity L01 p. 5 Sample space: Discrete example • Two rolls of a tetrahedral die – Sample space vs. sequential description Y = Second roll 1 2 3 4 1,1 1,2 1,3 1,4 4,4 L01 p. 6 Sample space: Continuous example " = {(x, y) | 0 ! x, y ! 1} x 1 1 y X = First roll 1 2 3 4 4 3 2 1 * *Athena is MIT's UNIX-based computing environment. OCW does not provide access to it.L01p.7 L01p.8 Probability axioms Probability law: Example with finite sample space • Event: a subset of the sample space • Probability is assigned to events Y = Second roll Axioms: 1. Nonnegativity: P(A) " 0 2. Normalization: P(")=1 X = First roll 1 2 3 4 4 3 2 1 3. Additivity: If A# B = Ø, then P(A$ B)= P(A)+ P(B) • Let every possible outcome have probability 1/16 • P({s1,s2,...,sk})= P({s1})+ ···+ P({sk}) – P((X,Y) is (1,1) or (1,2)) = = P(s1)+ + P(sk) – P({X =1})= ···Axiom 3 needs strengthening – P(X + Y is odd) = • • Do weird sets have probabilities? – P(min(X,Y) = 2) = L01 p. 9 L01 p. 10 Discrete uniform law Continuous uniform law • • Let all outcomes be equally likely Then, P(A)= number of elements of A total number of sample points • Two “random” numbers in [0,1]. 1 y • • Computing probabilities % counting Defines fair coins, fair dice, well-shuffled decks x1 • Uniform law: Probability = Area – P(X + Y ! 1/2) = ? – P((X,Y) = (0.5,0.3) ) L01 p. 11 Probability law: Ex. w/countably infinite sample space • Sample space: {1,2,...}– We are given P(n)=2−n , n =1,2,... – Find P(outcome is even) 1/2 ….. p 1/4 1/8 1/16 1 2 3 4 P({2,4,6,...})= P(2) + P(4) + ··· = 1 22 + 1 24 + 1 26 + ··· = 1 3 • Countable additivity axiom (needed for this calculation): If A1,A2,... are disjoint events, then: P(A1 $ A2 $ ···)= P(A1)+ P(A2)+ ··· MIT OpenCourseWarehttp://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied ProbabilityFall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.