Page 324 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 307

Fisher, R.A., 1925, Statistical Methods for Research Workers, 14th edn, Hafner, New
York, (1st edn, 1925).

Fisher, R.A., 1925, ‘‘Theory of Statistical Estimation’’, Proc. Camb. Phil. Soc. 22 700–
725.
Rao, C.R., 1945, ‘‘Information and Accuracy Attainable in the Estimation of Statistical
Parameters’’, Bull. Calcutta Math. Soc. 37 81–91.

PROBLEMS

The following notations and abbreviations are used in some statements of the problems:
X ˆ sample mean
x ˆ observed sample mean
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S ˆ sample variance
2
s ˆ observed sample variance
CRLB ˆ Crame ´r-Rao lower bound
ME ˆ moment estimator, or moment estimate
MLE ˆ maximum likelihood estimator, or maximum likelihood estimate
pdf ˆ probability density function
pmf ˆ probability mass function
9.1 In order to make enrollment projections, a survey is made of numbers of children in
100 families in a school district; the result is given in Table 9.2. Determine x, the
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observed sample mean, and s , the observed sample variance, on the basis of these
100 sample values.

Table 9.2 Data for Problem 9.1
Children (No.) Families (No.)
0 21
1 24
2 30
3 16
4 4
5 4
6 0
7 1
n ˆ 100

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9.2 Verify that the variance of sample variance S as defined by Equation (9.7) is given
by Equation (9.10).
9.3 Verify that the mean and variance of kth sample moment M k as defined by Equation
(9.14) are given by Equations (9.15).
9.4 Let X 1 , X 2 ,.. . , X 10 be a sample of size 10 from the standardized normal distribution
N(0, 1). Determine probability P(X 1).
9.5 Let X 1 , X 2 ,. . ., X 10 be a sample of size 10 from a uniformly distributed random
variable in interval (0, 1).

TLFeBOOK