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ISEN 609 Lecture 2

ISEN 609 Lecture 2 - Thinking about uncertainty food for...

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Thinking about uncertainty - food for thought: What is the likelihood that a coin is tossed and comes up heads? What is the likelihood that the proportion of heads in 5 tosses of a fair coin is 1/2? in 20 tosses? in 1,000,000 tosses?

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1. Probability Space (Random Experiment) and the Axioms of Probability Sample Space : Set of all possible outcomes of the experiment (list of all things that could possibly happen; only one of these actually will happen) Events Space : An event is a property that can be observed to happen (or not) after the experiment is performed; the event space is the collection of all events Probability Measure : Assigns a number to quantify the likelihood that any event happens a priori (“the probability of . ..”)
Sample Space - a set The set of all possible outcomes that could result when the random experiment is “performed”. Only one outcome will actually occur, but we don’t know which one in advance. Mathematically, we’ll think of each outcome as a point s , and the collection of outcomes as a set of points S . The sample space may be discrete; i.e., or continuous; i.e., Note: we can handle discrete sample spaces much more easily than continuous sample spaces S = { s 1 ,s 2 , . . . } { s S } S = { [0 , ) } or S = { [ a, b ) ( c, d ) }

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Sample Space - Examples Suppose I’m trying to build a model for coin tossing. What should be my sample space? How about a model for the roll of 2 dice? How about a model for jobs arriving at a machine for processing? In complicated experiments, it may be useful to think of elements of the sample space being represented by a “videotape” of a particular realization of the experiment
Event Space - a set The event space is a collection of subsets of the sample space with the following properties: The event space specifies what level of detail is observable about the outcomes; the larger the event space, the more detail is known about the outcome of the experiment Typically, many event spaces can be specified for a given sample space. We generally want to choose the

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ISEN 609 Lecture 2 - Thinking about uncertainty food for...

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