304_Hw8_Estimators - CEE 304 UNCERTAINTY ANALYSIS IN...

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CEE 304  -  UNCERTAINTY ANALYSIS IN ENGINEERING Homework #8 Due:   Monday, October 22, 2007.  Read:   Sections 6.1 and 6.2, and prob. 34 on p. 252 Devore6  [p. 279 Devore6]  for definition of  MSE. For confidence intervals please read 7.1-7.2. Goal:   We are now studying statistics, starting with the fundamental concept of  estimators.   The  purpose of this assignment is to give you an opportunity to work with and better understand the idea  of an estimator, and the origins of the method of moments and of maximum likelihood. Assignments ActivStats   Estimation Lesson  (optional this week) Look at the first two activities on page 18-2, “Know how to use a t distribution ...”  and “Learn to construct intervals …” (a) What is a 95% confidence interval for dissolved oxygen with n = 15?  Use the slider and read the interval from the graph in the second activity.  (b) Why do we need to use a Student t distribution instead of a normal distribution?   Devore and Other Problems 0.   This week the homework and reading took me: A B C D E 3-5 hrs 5-7 hrs 7-9 hrs 9-11 hrs > 12 hrs 1. Assume that you have collected lab test data {X 1 , .... , X n } and experience suggests that it  can be modeled by a Gamma distribution f X (x)  =  [ β α Γ(α29 ] -1   x α -1  exp( –x/ β ) with a  fixed  value of the shape parameter  α . Thus you only need to calculate the value  of  β  [In environmental applications when n is small,  α  may be computed as an average from many  samples to increase its precision.]   (a)  What is the maximum likelihood estimator of  β   in this case? (b)  What is the method of moments estimator of  β   in this case?   (c)  If  α were not known, then what non-linear equation would one need to  solve numerically to obtain the maximum likelihood estimator of  α ? How would  β  then be estimated?  [See also #27, Devore7, p. 251 {D6, p. 278}] 2.  Assume that the pdf for a random variable X is triangular: f(x)  =   (2/v) [ 1   –  (x/v) ] for 0   x   v,     and zero otherwise.   This is a simple probability model for a variable restricted to the interval [0, v].  
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