Consider a signal detection problem involving two hypotheses:Ho : X = WandwhereH₁ : X = 8 + W,Suppose we observeis a known signal vector, and Ŵ is a vector consisting of independent Gaussianrandom variables with mean 0 and variance 1. Suppose a priori that these twohypotheses are equally likely; that is,P(Ho) = P(H₁) =10S = -101Which hypothesis is more likely?0.630.21X = -1.190.84-0.04

Answered: Consider a signal detection problem…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Consider a signal detection problem involving two hypotheses: H₁ : X = W and where H₁ : X = š+ Ŵ, is a known signal vector, and W is a vector consisting of independent Gaussian random variables with mean 0 and variance 1. Suppose a priori that these two hypotheses are equally likely; that is P(H₁) = P(H₁) = 0.5. Suppose we observe X, but based on X we still determine that the two hypotheses are equally likely. What must be true about X?
Suppose that xi and x2 are random samples of observations from a population with mean m and variance s2. Consider the following three point estimators, A.B.C, of m: *1 + x2 *1 + 3r2 *1 + 2x2 C = A = B = 2 3 a) Which of these estimators yields an unbiased estimator of m? Justify your answer. b) Of those that are unbiased, which is the most efficient? Justify your answer.
Suppose we are interested in the effect of gender on income and we estimate a model in which income is the dependent variable and the regressor is a dummy variable that takes the value 1 if the person is a man and O if the person is a woman. If we instead estimate a model that includes a dummy variable for women (takes value 1 if woman and 0 if man), then The fit of the model will change. We will get a different conclusion for testing a null hypothesis that there is no gender effect. The coefficient on the dummy variables will differ across models, but the estimated difference of mean income between men and women will not change. The coefficient on the dummy variables will be exactly the same.
A possible important environmental determinant of lung function in children is the amount of cigarette smoking in the home. Suppose this question is studied by selecting two groups: Group 1 consists of 23 nonsmoking children 5-9 years of age, both of whose parents smoke, who have a mean forced expiratory volume (FEV) of 2.1 L and a standard deviation of 0.7 L; group 2 consists of 20 nonsmoking children of comparable age, neither of whose parents smoke, who have a mean FEV of 2.3 L and a standard deviation of 0.4 L.*8.31 What are the appropriate null and alternative hypotheses to compare the means of the two groups? *8.32 What is the appropriate test procedure for the hypotheses in Problem 8.31? *8.33 Carry out the test in Problem 8.32 using the criticalvalue method. *8.34 Provide a 95% CI for the true mean difference in FEV between 5- to 9-year-old children whose parents smoke and comparable children whose parents do not smoke. *8.35 Assuming this is regarded as a pilot study, how many…
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In the vast country of China lives one of the critically endangered species in the world, called the South China Tiger. Suppose enough jungle preserve is set aside so that there is suffcient room for more territories of this tiger specie than there are actual South China Tigers themselves. Hence, there will be no danger of overcrowding. But if the population is too small, the South China Tigers will have difficulty finding each other during mating season. 1. Identify the independent and dependent variables, and the parameters to be used. 2. Construct a population model based on the relevant assumptions. 3. Write a differential equation that models the population of South China Tigers.

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