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.,Xn fem a random sample from a distribution that belongs to an exponential famiy 0f
Assume that X 1, .
7.3. Prove that T = ELI d(X~i) is a sufcient statisti
Equivalance Between Tests of Hypothesis & Confidence Interval
In real situations problem of testing two simple hypotheses is rare.
Usually one of the two hypotheses is composite. The N-P lemma does not
In some situations, the theory can be expanded
4109 Fall 2012
In this class we reView
1 Unbiasedness and introduce
2 Convergence in probability
3 Convergence in distribution
4 the Central Limit Theorem Example (an unbiased estimator of 6) X; N U (U, 6), 9 unknown (want
to estimate), then the p.d.f. of
MVUE and sufcient Statistics
f (3L6) (Model based)
Let 6 be an unbiased estimator of 6,
19(6) : 6, Var(6) : E(6 6)2
It is desirable to nd an unbiased estimator of 6 with smallest variance.
Question: How do we nd an unbiased estimator of a parameter
The multiplication Principle
If one experiment has m outcomes and another experiment has n outcome,
then there are mn possible outcomes for the two experiments.
()1 . . . bn
In general if there are p experiments and the rst one has 711 possibl
Chapter 8. Confidence interval for the mean of N (, 2 )
Example Cell phone usage
A sample of 8 cell phone users revealed the following numbers (hours of
operation): 5, 3, 10, 13, 6, 1, 9, 7
Find a 95% confidence interval for the average hours of operation
',ii;1"_ _. i /'
Suppose that the number ofminutes required to serve a
customer at the checkout counter ofa supermarket has an
exponential distribution for which the mean is 3. Using the
central limit theorem, approximate the probability that
In summary, let T; be unbiased, E(T2)m 6, let T; be a. sufcient statistic
for 6. Then the random veeble Minx E(T2|T1) has E[go ($21)] = 6 and e.
Let X; be iid., f(33; 6),6 E 0
Let T; 2151(501, ~ we) be a. suicient statistic
Probability and Statistical Inference
Instructor: Professor Victor H. de la Pena, email@example.com
Probability and Statistics, by DeGroot and Schervish, Fourth Edition. Addison Wesley, NY.
We will cover the entire book in order
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