lecture3 - Stat 302, Introduction to Probability Jiahua...

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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 3 January-April 2011 1 / 56 Probability is a function on subsets of S Suppose we have a random experiment and identified its sample space S . The probability measure assigns a number between 0 and 1 to every subset of S . While each sample point is itself a subset of S , it has an associated probability only when it is regarded as a subset, not when it is sample point. Jiahua Chen () Lecture 3 January-April 2011 2 / 56 Example Random Experiments: randomly pick a number out of { 1, 2, 3 } . The sample space is S = { 1, 2, 3 } . S has 2 3 possible subsets: F = { , { 1 } , { 2 } , { 3 } , { 1, 2 } , { 1, 3 } , { 2, 3 } , { 1, 2, 3 }} . F is called -algebra. The probability has F as its domain, not S . Jiahua Chen () Lecture 3 January-April 2011 3 / 56 Example The generic element of S is denoted by s and called outcome or sample point. The generic element of F is denoted by E and called Event . In our example there are 3 outcomes and 2 3 = 8 events. For that reason, the collection of all possible subset of S , F , is often denoted by 2 S and called the power set of S . Jiahua Chen () Lecture 3 January-April 2011 4 / 56 The probability function, P ( ) , is defined on F = 2 S , not on S . that is P : 2 S [ 0, 1 ] . The domain of the probability function P is 2 S and its range is the closed interval [ 0, 1 ] . In other words, the probability function only takes as input subsets of S (events). Jiahua Chen () Lecture 3 January-April 2011 5 / 56 In our simple example, P ( { 1 } ) = P ( { 2 } ) = P ( { 3 } ) = 1 3 . Due to the axioms, we must make P ( ) = 0, P ( S ) = 1; P ( { 1, 2 } ) = P ( { 1, 3 } ) = P ( { 2, 3 } ) = 2 3 . Writing something like P ( 1 ) = 1 / 3 is technically incorrect because the sample point 1 is not an element of 2 S . The proper notation is P ( { 1 } ) = 1 / 3 because { 1 } is an element of 2 S . Jiahua Chen () Lecture 3 January-April 2011 6 / 56 Classical probabilities Axioms regulate what rules we must follow when assigning probabilities. They do not tell us how much probability we should assign to each single sample point event. Due to symmetry, many random experiments have finite and seemly equally likely outcomes. In these classical situations, we assign equal probabilities to every sample point (regarded as single sample point event). The resulting probability measure (function of event) P ( ) satisfies 3 Axioms when properly scaled. Consequently, probability calculations are often carried out through combinatorial analyses. Jiahua Chen () Lecture 3 January-April 2011 7 / 56 Combinatorial analyses aka Counting Assume a random experiment is under consideration....
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This note was uploaded on 04/07/2011 for the course STAT 302 taught by Professor Dr.chen during the Spring '11 term at The University of British Columbia.

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lecture3 - Stat 302, Introduction to Probability Jiahua...

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