Suppose that X1,...,Xn form a random sample from

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

Consider two p.d.f.s f0(x) and f1(x) that are defined as follows:

. 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

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

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and known variance is 1,

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

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

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

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 ()

. 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

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

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:

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