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# Lecture 9 - Lecture 9 Material Covered in This Lecture...

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Lecture 9 Material Covered in This Lecture: 1 Chapter 4, Section 4.4: Means and Variances of Random Variables;

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Mean (Expectation): Example 1 (Example 4.21, p.293): If the first digit in a set of data appear "at random", the nine possible digits 1 to 9 all have the same probability. The probability distribution of the first digit is then X 1 2 … … 9 P(X=x) 1/9 1/9 … … 1/9 The mean of the distribution of X is:
If, on the other hand, the data obey Benford's law, then what is the mean of X? 1 2 3 4 5 6

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7 8 9 10 Continuous case If X ~ f(x), then E(X)=∫ x f(x)dx Remark: If the distribution is symmetric, then the mean is equal to the symmetric center. 1 Rules For Means 2 Law of Large Numbers: As the number of observations, drawn randomly from a population X with finite mean μ X , increases, the sample mean X eventually approaches the mean μ X .

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Example 2 (Example 4.24, p.299): Linda is a sales associate at a large auto dealership. At her commission rate of 25% of gross profit on each vehicle she sells, Linda expects to earn \$350 for each car sold and \$400 for each

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