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# lecture6 - 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 6 January-April 2011 1 / 41 Average In a class of students, 30% of them get exactly 60, 50% of them get 80 and 20% of them get 95 in their final exam. What is the class average of this final exam? Apparently, the average is given by 0.30 60 + 0.50 80 + 0.20 95 = 77. Jiahua Chen () Lecture 6 January-April 2011 2 / 41 Average In a class of students, 30% of them get exactly 60, 50% of them get 80 and 20% of them get 95 in their final exam. What is the class average of this final exam? Apparently, the average is given by 0.30 60 + 0.50 80 + 0.20 95 = 77. Note that we did not use information on how many students this class has. Jiahua Chen () Lecture 6 January-April 2011 2 / 41 Average for a random variable In a class of students, 30% of them get exactly 60, 50% of them get 80 and 20% of them get 95 for their final marks. If I randomly select a student from this class, and let X be his/her final mark. What is the probability mass function of X ? Jiahua Chen () Lecture 6 January-April 2011 3 / 41 Average for a random variable In a class of students, 30% of them get exactly 60, 50% of them get 80 and 20% of them get 95 for their final marks. If I randomly select a student from this class, and let X be his/her final mark. What is the probability mass function of X ? The pmf of X is given by p ( 60 ) = 0.3; p ( 80 ) = 0.5; p ( 95 ) = 0.2. Jiahua Chen () Lecture 6 January-April 2011 3 / 41 Average for a random variable In a class of students, 30% of them get exactly 60, 50% of them get 80 and 20% of them get 95 for their final marks. If I randomly select a student from this class, and let X be his/her final mark. What is the probability mass function of X ? The pmf of X is given by p ( 60 ) = 0.3; p ( 80 ) = 0.5; p ( 95 ) = 0.2. In view of these information, what is the average mark of a randomly selected student? Jiahua Chen () Lecture 6 January-April 2011 3 / 41 Average for a random variable While the outcome of X is random due to the randomness in selecting the student, the proportion of times when X = 60 is clearly 30% if we have it repeated INFINITE many times. Similarly, 50% of such X values are 80, and 20% such values are 95. Thus, it is sensible to use the average outcomes over INFINITE repetitions as the average size of X . Let us tentatively define the Expectation of this X as E ( X ) = 0.30 60 + 0.50 80 + 0.20 95 = 77. Jiahua Chen () Lecture 6 January-April 2011 4 / 41 Average for a random variable Suppose X is a discrete random variable and its range is given by { x 1 , x 2 , . . . , } , and its pmf is given by p ( x ) . We define the expectation of X as E ( X ) = ?...
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## lecture6 - Stat 302, Introduction to Probability Jiahua...

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