Here we consider a random variable G whose probability density function (PDF) depends on a parameter,...

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Here we consider a random variable G whose probability density function (PDF) depends on a parameter, call it H. We start by modeling this problem under the assumption that H is random with a given PDF. In this case, it makes sense to think of the PDF for G as being conditioned on H = h. To be concrete, consider the case where the two PDFs are a fGLH (glb) == {1/2 0sgsh 1/2 hsgs1 else (1/4 Oshsb fu(h)=7/4 bshsc 0 else with constants a, b, and c. 1. Determine the constant a. 2. Show that b = 1/2, and c = 1 if we know that E[H] = 11/16. 3. Determine the joint distribution for G and H. 4. Determine the marginal distribution for G. Now we consider the problem where somehow we know H is either 1/4 or 3/4 and must make a choice as to which based on data. 5. Given a value for G, what is the maximum likelihood rule for determining whether H is 1/4 or 3/4? 6. Suppose now that we are given a series of independent and identically distributed samples G; for i= 1,2,..., N. What is the maximum likelihood rule for choosing between the two values for H based on the number of times G; > .5 from the N samples? Simplify your results as much as possible. Here we consider a random variable G whose probability density function (PDF) depends on a parameter, call it H. We start by modeling this problem under the assumption that H is random with a given PDF. In this case, it makes sense to think of the PDF for G as being conditioned on H = h. To be concrete, consider the case where the two PDFs are a fGLH (glb) == {1/2 0sgsh 1/2 hsgs1 else (1/4 Oshsb fu(h)=7/4 bshsc 0 else with constants a, b, and c. 1. Determine the constant a. 2. Show that b = 1/2, and c = 1 if we know that E[H] = 11/16. 3. Determine the joint distribution for G and H. 4. Determine the marginal distribution for G. Now we consider the problem where somehow we know H is either 1/4 or 3/4 and must make a choice as to which based on data. 5. Given a value for G, what is the maximum likelihood rule for determining whether H is 1/4 or 3/4? 6. Suppose now that we are given a series of independent and identically distributed samples G; for i= 1,2,..., N. What is the maximum likelihood rule for choosing between the two values for H based on the number of times G; > .5 from the N samples? Simplify your results as much as possible.

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