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probability_primer

# probability_primer - Spring 2005 Economics 425 John Rust...

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Spring 2005 John Rust Economics 425 University of Maryland Primer on Probability, Random Variables, and Stochastic Processes 1 Basic Probability Theory These notes provide a basic introduction to probability theory. Having a rudimentary understanding of probability theory, particularly the topic of random variables, is essential to be able to understand the economic topics of 1) choice under uncertainty, and 2) dynamic programming, that are covered in Econ 425. Definition 1: A probability space ( P , Ω , F ) is a mapping P : F from a sigma algebra F , a collection of subsets of the sample space Ω , to the [ 0 , 1 ] interval. The sample space Ω can be viewed as the universe of all possible events, or “states of nature” and the sigma algebra F can be viewed as a collection of events (subsets of possible states of nature). The sigma algebra F has to satisfy the following properties: S1. Ω F S2. /0 F (i.e. the empty set, /0 is in F ) S3. if A is a member of F , then its complement, A c , must also be a member of F (i.e. we can write this statement mathematically as A F = A c F ). S4. if A F and B F , then A B F and A B F (i.e. if A and B are events in F , the union and intersection of these sets are also events in F ), S5. if { A i } is a possibly infinite collection of events with each A i F , then i = 1 A i F and i = 1 A i F (i.e. the union and complement of an infinite sequence of events in F is also in F ). The probability function P : F [ 0 , 1 ] must satisfy P1. P ( Ω ) = 1 P2. P ( /0 ) = 0 P3. if A and B are in F and A B = /0 , then P ( A B ) = P ( A )+ P ( B ) P4. if A and B are in F and A B = /0 , then P ( A B ) P ( A )+ P ( B ) P5. if { A i } is an infinite sequence of events in F , then P ( i = 1 ) i = 1 P ( A i ) Thus, P ( A ) can be enterpreted as the probability of the event A occuring, and P ( A B ) is the prob- ability of the event “ A or B ” and P ( A B ) is the probability of the event “ A and B ” and P ( A c ) is the probability of the event “not A ” (i.e. the event that event A did not occur). We let the singleton element ω Ω denote the actual state of the world. Since the world is very complex (i.e. the state of the world might involve describing the positions, masses, charges/energies, and momentum of every particle in the universe) we typically do not directly observe the actual state of the world ω . However we can tell whether certain events occurs, which are sets of states of nature that all result in the same (observable) 1

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outcome. Thus, if ω A (i.e. event A contains the true state of nature ω ), then this is what we mean by saying, “event A has occurred”. An example event A might be “it is raining in College Park at noon Saturday, March 12, 2005”. There are many different states of the world (as described by the exact spec- ification of all atoms and molecules in the universe) that are consistent with event A occuring. However as long any one ω occurs so that it rains in College Park at noon on March 12, 2005, then event A has occurred.
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