ch10_prob

ch10_prob - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . Harvard-MIT Division of Health Sciences and Technology HST.582J: Biomedical Signal and Image Processing, Spring 2007 Course Director: Dr. Julie Greenberg HST-582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2007 Chapter 10 - A PROBABILITY PRIMER CDFs, PMFs, PDFs, Expectation and all that... c J.W. Fisher 2007 10 Introduction In many experiments there is some element of randomness that we are unable to explain. Proba- bility and statistics are mathematical tools for reasoning in the face of such uncertainty. Here, we are primarily interested in the use of probability for decision making, estimation, and cost min- imization. Probabilistic models allow us to address such issues quantitatively. For example; “Is the signal present or not?” Binary : YES or NO, “How certain am I?” Continuous : Degree of confidence. Here we introduce some very basic notions of probability and random variables and their associated cumulative distribution functions (CDF), probability mass functions (PMF), and probability density functions (PDF). From these we can define the notion of expectation, marginal- ization, and conditioning. This description is meant to be concise, limiting the discussion to those concepts which can be applied to decision problems we encounter in biomedical signal and image processing 1 . 10.1 Sample Spaces and Events Basic probability can be derived from set theory where one considers a sample space , associated events , and a probabilty law . The sample space , denoted Ω , is the exhaustive set of finest grain outcomes of an experiment. An event is a subset of the sample space . A probability law , denoted P , assigns numerical values to events . While the sample space may be continuous or discrete, we restrict the following discussion to discrete (countable) sets in order introduce basic probability concepts. Examples of sample spaces are: • the set of outcomes of rolling a 6-sided die, • the set of outcomes of rolling a 6 sided die AND flipping a coin, • the set of outcomes when drawing 3 (or 4) cards from 47 Examples of events for the above sample spaces are: • the roll of the die is greater than 4, 1 more accurately, we consider concepts which can be applied to a limited, but useful class of decision problems. 1 Cite as: J.W. Fisher. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology....
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This note was uploaded on 11/07/2011 for the course AERO 16.422 taught by Professor Juliegreenberg during the Spring '07 term at MIT.

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ch10_prob - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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