STAT_340_Course_Notes

STAT_340_Course_Notes - Stat 340 Course Notes Spring 2010...

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Unformatted text preview: Stat 340 Course Notes Spring 2010 Generously Funded by MEF Contributions: Riley Metzger Michaelangelo Finistauri Special Thanks: Without the following people and groups, these course notes would never have been completed: MEF, Don McLeish. Chapter 1 Probability Three approaches to de&ning probability are: 1. The classical de&nition: Let the sample space (denoted by S ) be the set of all possible distinct outcomes to an experiment. The probability of some event is number of ways the event can occur number of outcomes in S ; provided all points in S are equally likely. For example, when a die is rolled, the probability of getting a 2 is 1 6 because one of the six faces is a 2. 2. The relative frequency de&nition: The probability of an event is the proportion (or fraction) of times the event occurs over a very long (the-oretically in&nite) series of repetitions of an experiment or process. For example, this de&nition could be used to argue that the probability of get-ting a 2 from a rolled die is 1 6 . For instance, if we roll the die 100 times, but get a 2 30 times, we may suspect that the probability of getting a 2 is 1 3 . 3. The subjective probability de&nition: The probability of an event is a measure of how sure the person making the statement is that the event will happen. For example, after considering all available data, a weather forecaster might say that the probability of rain today is 30% or 0.3. Unfortunately, all three of these de&nitions have serious limitations. 1 1.1 Sample Spaces and Probability Consider a phenomenon or process which is repeatable, at least in theory, and suppose that certain events (outcomes) A 1 ;A 2 ;A 3 ;::: are de&ned. We will often refer to this phenomenon or process an &experiment," and refer to a single repetition of the experiment as a &trial . " Then, the probability of an event A , denoted by P ( A ) , is a number between 0 and 1. Denition 1. A sample space S is a set of distinct outcomes for an exper-iment or process, with the property that in a single trial, one and only one of these outcomes occurs. The outcomes that make up the sample space are called sample points . Denition 2. Let S = f A 1 ;A 2 ;A 3 ;::: g be a discrete sample space. Then probabilities P ( A i ) are numbers attached to the A i &s ( i = 1 ; 2 ; 3 ;::: ) such that the following two conditions hold: (1) & P ( A i ) & 1 (2) P i P ( A i ) = 1 The set of values f P ( A i ) ; i = 1 ; 2 ;::: g is called a probability distrib-ution on S . Denition 3. An event in a discrete sample space is a subset A S . If the event contains only one point, e.g. A 1 = f A 1 g , we call it a simple event. An event A made up of two or more simple events such as A = f A 1 ;A 2 g is called a compound event....
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STAT_340_Course_Notes - Stat 340 Course Notes Spring 2010...

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