qitd311 - Probabilities Robert B. Griffiths Version of 12...

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Probabilities Robert B. Grifths Version oF 12 January 2010 ReFerences: ±eller, An introduction to probability theory and its applications , Vol. 1, 3d ed (Wiley 1968). See Intro- duction, Ch. I, Ch. V DeGroot and Schervish, Probability and Statistics , 3d ed (Addison-Wesley, 2002), Chs 1, 2, 3, 4 Very compact introductions to material relevant to quantum mechanics: QCQI = Quantum Computation and Quantum Information by Nielsen and Chuang (Cambridge, 2000), App. 1 CQT = Consistent Quantum Theory by Robert Grifths (Cambridge, 2002), Secs. 5.1, 8.2, 9.1, 9.2 Contents 1B a s i c s 1 1.1 Sample space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 1 1.2 Event algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 2 1.3 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 2 1.4 Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 3 1.5 Conditional probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... 3 2R a n d omV a r i a b l e s 3 2.1 Random variables, averages, indicator Functions . . . . . . . . . . . . . . . . . ......... 3 2.2 Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... 4 2.3 Joint and conditional distributions . . . . . . . . . . . . . . . . . . . . . . . . ......... 4 2.4 Independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... 5 2.5 Variance, covariance, correlation . . . . . . . . . . . . . . . . . . . . . . . . . ......... 6 3 Stochastic Processes 6 3.1 Examples; sample spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... 6 3.2 Probabilities . 7 3.3 Markov chains 7 1 Basics ± Ordinary (classical) probability theory uses three key concepts: sample space S , event algebra B , probabilities P . 1.1 Sample space ± Sample space S oF mutually exclusive possibilities , thought oF as outcomes oF an ideal experiment, one and only one oF which actually occurs, or is true, in a particular case. Examples. Coin toss: H or T . Die: { s =1 , 2 , 3 , 4 , 5 , 6 } A sample space can be either discrete or continuous (e.g., integers, real numbers), and in the Former case either ²nite or in²nite. ±or our purposes it sufces to consider ²nite discrete sample spaces. 1
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1.2 Event algebra ± Event algebra B consisting of a collection of subsets of S . In the simplest situation it is the collection of all subsets of S , but is sometimes a smaller collection. The collection B must be closed under the operations of union, intersection, and complement, and must contain S as one of its elements. Elements of B are known as “events.” The symbol B is used because this is a Boolean algebra under the usual set-theoretic operations. Example: For a coin there are 2 2 = 4 possible subsets of the set { H,T } , namely (the empty set; yes it must be included in the algebra), { H } , { T } , { } . We won’t worry too much about the di±erence between T and { T } . More interesting example. For a die, s =6isa simple event, corresponding to a single element from the sample space, while s 3 is an example of a compound event, as it corresponds to the subset { 3 , 4 , 5 , 6 } . Another compound event: s is even. The adjectives “simple” and “compound” are not always used, but are sometimes convenient. ± Exercise. How big is the algebra containing all subsets of S = { 1 , 2 , 3 , 4 , 5 , 6 } in the case of a die? A case where B does not consist of all subsets: We are only interested in whether the number of spots on the die is even or odd. One event in B is { 1 , 3 , 5 } , a second is { 2 , 4 , 6 } , and B contains their intersection and their union S , so four subsets in all.
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This note was uploaded on 02/15/2012 for the course PHYS 3101 taught by Professor Staff during the Spring '08 term at Pittsburgh.

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qitd311 - Probabilities Robert B. Griffiths Version of 12...

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