21notes4.pdf - Notes 21 Desired outcomes from last class...

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Notes 21 Desired outcomes from last class Students will be able to: derive the moment generating function of a r.v.; find E ( X r ) for an arbitrary r using the m.g.f. of X . 8 Limit Theorems Markov’s Inequality: For P ( X 0 ) = 1 , a > 0 , P ( X a ) 1 a E ( X ) . Chebyshev’s inequality: (Two alternative forms) P ( | X - μ | ≥ k σ ) 1 k 2 , and P ( | X - μ | ≥ ) σ 2 2 . Notes 21 8 Limit Theorems Example 2a on page 368: X is the number of items produced in a factory during a week. Let μ = E ( X ) = 50 . (a) What can be said about the probability that this week’s production will exceed 75? (b) If the variance of a week’s production is known to equal 25 , then what can be said about the probability that this week’s production will be between 40 and 60? Notes 21 8 Limit Theorems Problem 8.2 on page 390: From past experience a professor knows that the test score of a student taking the final examination is a random variable with mean 75. (a) Give an upper bound for the probability that a student’s test score will exceed 85. (b) Suppose, in addition, that the professor knows that the variance of a student’s test score is equal to 25. What can be said about the probability that a student will score between 65 and 85? (c) How many students would have to take the examination to ensure with probability at least 0.9 that the class average would be within 5 of 75?
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Notes 21 8 Limit Theorems Problem 8.4a on page 390: X 1 , . . . , X 20 independent Poisson r.v. with mean 1. Use, Markov inequality to obtain bound on P 20 i = 1 X i > 15 .
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