- #31
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For part C, I suggest the easiest way to get the overall density function is to define ##A = \frac{\min\{\Phi, \Psi\}}{\Theta}## and ##B = \frac{\max\{\Phi, \Psi\}}{\Theta}##. Then consider three ranges: A+B < 1, B-A < 1 < B+A, B-A > 1. You should get a fairly simple graph.
I don't see this as a step towards part D, though. For part D, you have a sequence of scores Xi out of N. You could try an MLE approach, but it gets horrendous. You'd need a separate ##\psi_i## parameter for each Xi, and maximise likelihoods based on each shot having success probability ##\Theta p_i = \min\{\Theta, \psi_i+\phi\} - \max\{-\Theta, \psi_i-\phi\}##; even then that's only if ##\psi_i-\phi < \Theta## and ##\phi-\psi_i > - \Theta##; outside that range it's zero. And the likelihood of Xi is a binomial function of pi.
So I suggest putting all the algebra to one side and approaching D in a more commonsense manner. If ##\Psi## is large and ##\Phi## is small, what would you expect the distribution of Xi to look like? What about small ##\Psi## and large ##\Phi## ?
I don't see this as a step towards part D, though. For part D, you have a sequence of scores Xi out of N. You could try an MLE approach, but it gets horrendous. You'd need a separate ##\psi_i## parameter for each Xi, and maximise likelihoods based on each shot having success probability ##\Theta p_i = \min\{\Theta, \psi_i+\phi\} - \max\{-\Theta, \psi_i-\phi\}##; even then that's only if ##\psi_i-\phi < \Theta## and ##\phi-\psi_i > - \Theta##; outside that range it's zero. And the likelihood of Xi is a binomial function of pi.
So I suggest putting all the algebra to one side and approaching D in a more commonsense manner. If ##\Psi## is large and ##\Phi## is small, what would you expect the distribution of Xi to look like? What about small ##\Psi## and large ##\Phi## ?