practice01_3sol

practice01_3sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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Department of Civil and Environmental Engineering 1.017 Computing and Data Analysis for Environmental Applications Practice Quiz 3 December 5, 2001 Please answer all questions on a separate piece(s) of paper with your name clearly identified: Problem 1 (15 points) Answer each of the following questions in a few sentences: a) Suppose that y is a random variable (e.g nitrate concentration in a lake). Also, suppose that you have taken 20 nitrate measurements y 1 , y 2 , …, y 20 at various times and locations. Explain the difference between E [ y ], the expectation of y , and m y , the sample mean of y . b) Define the concept of a random sample and explain why it is useful c) Why is the sample mean of y 1 , y 2 , …, y 20 a “good’ estimate of E [ y ]? Solution: a) The random variable y is characterized by its probability density, which describes the likelihood of obtaining an observation in a given range (or interval) of values. The expectation E [ y ] is the first moment of this density. It is a property of the density rather than any particular set of data. The sample mean m y is the arithmetic average of a particular set of data and is derived from the data rather than the density. b) A sample is a set of measurements y 1 , y 2 , …, y n of some random variable y with a probability density f y ( y ). The n measured values can be viewed as particular outcomes associated with n related random variables, one for each member of the sample. Generally speaking, these n random variables are described by a multivariate probability distribution. If the sample is random, the y i ’s are independent and all have the same marginal probability density f y ( y ) as the original variable y . These properties greatly simplifies the task of deriving the probability distributions of estimates computed from the sample. c) The sample mean is a good estimate of E [ y ] because it is unbiased [ E ( m y )= E ( y )] and consistent [ Var ( m y ) goes to zero as the sample size approaches infinity]. Together, these properties imply that the sample mean converges to the expected value of y as the sample size increases. 1
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practice01_3sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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