Introduction to Probability and Statistics for Engineers and Scientists

278 Chapter 7: Parameter Estimation

2. Determine the maximum likelihood estimator ofθwhenX 1 ,...,Xnis a sample
   with density function

f(x)=^12 e−|x−θ|, −∞<x<∞

3. LetX 1 ,...,Xnbe a sample from a normalμ,σ^2 population. Determine the max-
   imum likelihood estimator ofσ^2 whenμis known. What is the expected value of
   this estimator?


4. The height of a radio tower is to be measured by measuring both the horizontal
   distanceXfrom the center of its base to a measuring instrument and the vertical
   angle of the measuring device (see the following figure). If five measurements of
   the distanceLgive (in feet) values

150.42, 150.45, 150.49, 150.52, 150.40

and four measurements of the angleθgive (in degrees) values

40.26, 40.27, 40.29, 40.26

estimate the height of the tower.

tower

X

q

5. Suppose thatX 1 ,...,Xnare normal with meanμ 1 ;Y 1 ,...,Ynare normal with
   meanμ 2 ; andW 1 ,...,Wnare normal with meanμ 1 +μ 2. Assuming that all 3n
   random variables are independent with a common variance, find the maximum
   likelihood estimators ofμ 1 andμ 2.


6. River floods are often measured by their discharges (in units of feet cubed per
   second). The valuevis said to be the value of a 100-year flood if

P{D≥v}=.01

whereDis the discharge of the largest flood in a randomly chosen year. The

following table gives the flood discharges of the largest floods of the Blackstone

River in Woonsocket, Rhode Island, in each of the years from 1929 to 1965.

Assuming that these discharges follow a lognormal distribution, estimate the value

of a 100-year flood.