Let f0(x) be the p.f. of the Bernoulli distribution with parameter 0.3, and let f1(x) be the p.f. of the Bernoulli distribution with parameter 0.6. Suppose that a single observation X is taken from a distribution for which the p.d.f. f (x) is either f0(x) or f1(x), and the following simple hypotheses are to be tested: H0: f (x) = f0(x), H1: f (x) = f1(x). Find the test procedure for wh
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1.10
Introduction to Probability
1.12
Introduction to Probability
1.4
Introduction to Probability
1.5
Introduction to Probability
1.6
Introduction to Probability
1.7
Introduction to Probability
1.8
Introduction to Probability
1.9
Introduction to Probability
2.1
Conditional Probability
2.2
Conditional Probability
2.3
Conditional Probability
2.4
Conditional Probability
2.5
Conditional Probability
3.1
Random Variables and Distributions
3.10
Random Variables and Distributions
3.11
Random Variables and Distributions
3.2
Random Variables and Distributions
3.3
Random Variables and Distributions
3.4
Random Variables and Distributions
3.5
Random Variables and Distributions
3.6
Random Variables and Distributions
3.7
Random Variables and Distributions
3.8
Random Variables and Distributions
3.9
Random Variables and Distributions
4.1
Expectation
4.2
Expectation
4.3
Expectation
4.4
Expectation
4.5
Expectation
4.6
Expectation
4.7
Expectation
4.8
Expectation
4.9
Expectation
5.10
Special Distributions
5.11
Special Distributions
5.2
Special Distributions
5.3
Special Distributions
5.4
Special Distributions
5.5
Special Distributions
5.6
Special Distributions
5.7
Special Distributions
5.8
Special Distributions
5.9
Special Distributions
6.1
Large Random Samples
6.2
Large Random Samples
6.3
Large Random Samples
6.4
Large Random Samples
6.5
Large Random Samples
7.1
Estimation
7.10
Estimation
7.2
Estimation
7.3
Estimation
7.4
Estimation
7.5
Estimation
7.6
Estimation
7.7
Estimation
7.8
Estimation
7.9
Estimation
8.1
Sampling Distributions of Estimators
8.2
Sampling Distributions of Estimators
8.3
Sampling Distributions of Estimators
8.4
Sampling Distributions of Estimators
8.5
Sampling Distributions of Estimators
8.6
Sampling Distributions of Estimators
8.7
Sampling Distributions of Estimators
8.8
Sampling Distributions of Estimators
8.9
Sampling Distributions of Estimators
9.1
Testing Hypotheses
9.10
Testing Hypotheses
9.2
Testing Hypotheses
9.3
Testing Hypotheses
9.4
Testing Hypotheses
9.5
Testing Hypotheses
9.6
Testing Hypotheses
9.7
Testing Hypotheses
9.8
Testing Hypotheses
9.9
Testing Hypotheses
10.1
Categorical Data and Nonparametric Methods
10.2
Categorical Data and Nonparametric Methods
10.3
Categorical Data and Nonparametric Methods
10.4
Categorical Data and Nonparametric Methods
10.5
Categorical Data and Nonparametric Methods
10.6
Categorical Data and Nonparametric Methods
10.7
Categorical Data and Nonparametric Methods
10.8
Categorical Data and Nonparametric Methods
10.9
Categorical Data and Nonparametric Methods
11.1
Linear Statistical Models
11.2
Linear Statistical Models
11.3
Linear Statistical Models
11.4
Linear Statistical Models
11.5
Linear Statistical Models
11.6
Linear Statistical Models
11.7
Linear Statistical Models
11.8
Linear Statistical Models
11.9
Linear Statistical Models
12.1
Simulation
12.2
Simulation
12.3
Simulation
12.4
Simulation
12.5
Simulation
12.6
Simulation
12.7
Simulation
Textbook Solutions for Probability and Statistics
Chapter 9.2 Problem 8
Question
Suppose that a single observation X is taken from the uniform distribution on the interval \([0,\ \theta]\), where the value of \(\theta\) is unknown, and the following simple hypotheses are to be tested:
\(\begin{array}{ll}H_0:&\theta=1,\\ H_1:&\theta=2\end{array}\)
a. Show that there exists a test procedure for which \(\alpha(\delta)=0\) and \(\beta(\delta)<1\)
b. Among all test procedures for which \(\alpha(\delta)=0\), find the one for which \(\beta(\delta)\) is a minimum.
Solution
Step 1 of 3
(a) Consider the following test \(\delta\) which rejects the null if and only if X > 1. Then,
\(\alpha(\delta)=P_{H_0}(X>1)=P(X>1\mid X\sim U(0,\ 1))=0\)
and
\(\beta(\delta)=P_{H_1}(X>1)=P(X>1\mid X\sim U(0,\ 2))=\frac{1}{2}<1\)
Thus this test \(\delta\) satisfies our required condition.
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full solution
full solution
Title
Probability and Statistics 4
Author
Morris H. DeGroot, Mark J. Schervish
ISBN
9780321500465
Solved: Suppose that a single observation X is taken from
Chapter 9.2 textbook questions
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Chapter 9: Problem 1 Probability and Statistics 4
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Chapter 9: Problem 2 Probability and Statistics 4
Consider two p.d.f.s f0(x) and f1(x) that are defined as follows:
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Chapter 9: Problem 3 Probability and Statistics 4
. Consider again the conditions of Exercise 2, but suppose now that it is desired to find a test procedure for which the value of 3() + () is a minimum. a. Describe the procedure. b. Determine the minimum value of 3() + () attained by
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Chapter 9: Problem 4 Probability and Statistics 4
Consider again the conditions of Exercise 2, but suppose now that it is desired to find a test procedure for which () 0.1 and () is a minimum. a. Describe the procedure. b. Determine the minimum value of () attained by the proced
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Chapter 9: Problem 5 Probability and Statistics 4
Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and known variance is 1,
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Chapter 9: Problem 6 Probability and Statistics 4
Suppose that X1,...,Xn form a random sample from the Bernoulli distribution with unknown parameter p. Let p0 and p1 be specified values such that 0 < p1 < p0 < 1, and suppose th
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Chapter 9: Problem 7 Probability and Statistics 4
Suppose that X1,...,Xn form a random sample from the normal distribution with known mean and unknown variance 2, and the following simple hypotheses are to be tested: H0: 2 = 2, H1: 2 = 3. a. Show that am
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Chapter 9: Problem 8 Probability and Statistics 4
Suppose that a single observation X is taken from the uniform distribution on the interval [0, ], where the value of is unknown, and the following simple hypotheses are to be tested: H0: = 1, H1: = 2. a. Show that th
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Chapter 9: Problem 9 Probability and Statistics 4
Suppose that a random sample X1,...,Xn is drawn from the uniform distribution on the interval [0, ], and consider again the problem of testing the simple hypotheses described in Exercise 8. Find the minimum value of () that can be attained among all test procedures for which ()
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Chapter 9: Problem 10 Probability and Statistics 4
. Suppose that X1,...,Xn form a random sample from the Poisson distribution with unknown mean . Let 0 and 1 be specified values such that 1 > 0 > 0, and suppose that it is desire
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Chapter 9: Problem 11 Probability and Statistics 4
1. Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and known standard deviation 2, and the following simple hypotheses are to be tested: H0: = 1, H1: = 1. Determine the minimum va
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Chapter 9: Problem 12 Probability and Statistics 4
Let X1,...,Xn be a random sample from the exponential distribution with unknown parameter . Let 0 < 0 < 1 be two possible values of the parameter. Suppose that we wish to test the following hypotheses: H0: = 0, H1:
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Chapter 9: Problem 13 Probability and Statistics 4
Consider the series of examples in this section concerning service times in a queue. Suppose that the manager observes two service times X1 and X2. It is easy to see that both f1(x) in (9.2.1) and f0(x) in (9.2.2) depend on the observed data only through the value t = x1 + x2 of the statistic T = X1 + X2. Hence, the tests from Theorems 9.2.1 and 9.2.2 both depend only on the value of T . a. Using Theorem 9.2
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