Name(s):
:
Estimators Break the Ice Problems
Tom Linton, Due March 29, 2000
-
The first bus of the day arrives at 46th and Grand Ave. at X minutes after
6 AM. Assume that X is uniformly distributed from 0 to b minutes (continuously).
The arrival times (in minutes after 6 AM) of the first bus of the day,
for one week were:
| X1 |
X2 |
X3 |
X4 |
X5 |
X6 |
X7 |
| 5.162 |
2.756 |
4.149 |
0.969 |
1.773 |
7.392 |
5.211 |
Let Xmin = Min{X1, X2, X3, X4, X5, X6, X7} and Xmax = Max{X1,
X2, X3, X4, X5, X6, X7}.
-
Calculate the values of each of the following estimators of the parameter
b:
-
=
2
.
-
=
Xmax.
-
=
Xmin + Xmax.
-
Decide if each of the above estimators is biased or unbiased. For each
biased estimator, adjust the estimate slightly (try multiplying it by something
or adding something to it) so that it becomes unbiased. You may wish to
use Mathematica's Integrate command here.
-
Calculate the variance of the first two (adjusted) estimators above. Which
is a better estimator of the parameter b?
-
Assume that X is a continuous random variable with density function f(x)
= e- (x - k), for x >= k (x bigger than or equal to k), and
that X1, X2, X3, X4, X5
is a random sample from the population X.
-
Show that
=
- 1 is an unbiased estimator of the parameter k.
-
Calculate the variance, V()
of this estimator.