probability_primer

Probability_primer - Spring 2005 Economics 425 John Rust University of Maryland Primer on Probability Random Variables and Stochastic Processes 1

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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. Defnition 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, 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 inFnite 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 inFnite 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 P3. if A and B are in F and A B = , then P ( A B ) = P ( A )+ P ( B ) P4. if A and B are in F and A B 6 = , then P ( A B ) P ( A P ( B ) P5. if { A i } is an inFnite 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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- iFcation 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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/25/2011 for the course ECON 326 taught by Professor Hulten during the Spring '08 term at Maryland.

Page1 / 31

Probability_primer - Spring 2005 Economics 425 John Rust University of Maryland Primer on Probability Random Variables and Stochastic Processes 1

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online