Page 329 - Fundamentals of Probability and Statistics for Engineers

312 Fundamentals of Probability and Statistics for Engineers

Let T 1 , T 2 ,... , T n be a sample from T.
^
^
(a) Determine the MLE and ME for ML and ME , respectively) assuming t 0 is
known.
^
^
(b) Determine the MLE and ME for t 0 (T OML and T OME , respectively) assuming
is known.
(c) Determine the MLEs and MEs for both and t 0 assuming both are unknown.
9.27 If X 1 , X 2 , ..., X n is a sample from the gamma distribution; that is,
r r 1
x
f x; r; †ˆ e x ; x 0; r; > 0;

r†
show that:
(a) If r is known and is the parameter to be estimated, both the MLE and ME
^
for are ˆ r/X.
(b) If both r and are to be estimated, then the method of moments and the

method of maximum likelihood lead to different estimators for r and . (It is
not necessary to determine these estimators.)
9.28 Consider the Buffalo yearly snowfall data, given in Problem 8.2(g) (see Table 8.6)
and assume that a normal distribution is appropriate.
(a) Find estimates for the parameters by means of the moment method and the
method of maximum likelihood.
(b) Estimate from the model the probability of having another blizzard of 1977
[P(X > 199.4)].
9.29 Recorded annual flow Y (in cfs) of a river at a given point are 141, 146, 166, 209, 228,
234, 260, 278, 319, 351, 383, 500, 522, 589, 696, 833, 888, 1173, 1200, 1258, 1340,
1390, 1420, 1423, 1443, 1561, 1650, 1810, 2004, 2013, 2016, 2080, 2090, 2143, 2185,
2316, 2582, 3050, 3186, 3222, 3660, 3799, 3824, 4099, and 6634. Assuming that Y
follows a lognormal distribution, determine the MLEs of the distribution parameters.
9.30 Let X 1 and X 2 be a sample of size 2 from a uniform distribution with pdf
8
1
< ; for 0 x ;
f x; †ˆ
0; elsewhere.
:
Determine constant c so that the interval
0 < < c X 1 ‡ X 2 †
is a [100(1 )]% confidence interval for .
9.31 The fuel consumption of a certain type of vehicle is approximately normal, with
standard deviation 3 miles per gallon. If a sample of 64 vehicles has an average fuel
consumption of 16 miles per gallon:
(a) Determine a 95% confidence interval for the mean fuel consumption of all
vehicles of this type.
(b) With 95% confidence, what is the possible error if the mean fuel consumption
is taken to be 16 miles per gallon?
(c) How large a sample is needed if we wish to be 95% confident that the mean will
be within 0.5 miles per gallon of the true mean?

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