Fieller-Creasy problem The simplest version N(2, 1), involves a pair of independent random variables X ~N(0,...

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Fieller-Creasy problem The simplest version N(2, 1), involves a pair of independent random variables X ~N(0, 1) and X2~ where both 6, and 2 are unknown. The quantity of interest is the ratio = 02/01- This is a famous (or infamous) example because weird things can happen. (a) Define Qo(X)= X2-6X1 (1+62)1/21 = where X (X1, X2), and show that Q.(X) has a standard normal distribution, i.e., it's a pivot. (b) Define a 100(1 - a)% confidence set for as Ca(1)={\Q(2) 2), where = a/2- Rewrite the inside inequality as "Q(x)22" and show that it can be reduced to a quadratic inequality A+B+C 0, where the coefficients depend on data and cutoff z but not on p. = (c) The confidence set is unbounded if the coefficient A A(X, 2) is negative. Find the probability2 P(A < 0) as a function of 8. That probability only depends on 6, so plot it over the range 6, E [-3,3]; with z= 1.96. For what values of 6, is that probability relatively high? Give an intuitive explanation for why those 0, values might be problematic. Fieller-Creasy problem The simplest version N(2, 1), involves a pair of independent random variables X ~N(0, 1) and X2~ where both 6, and 2 are unknown. The quantity of interest is the ratio = 02/01- This is a famous (or infamous) example because weird things can happen. (a) Define Qo(X)= X2-6X1 (1+62)1/21 = where X (X1, X2), and show that Q.(X) has a standard normal distribution, i.e., it's a pivot. (b) Define a 100(1 - a)% confidence set for as Ca(1)={\Q(2) 2), where = a/2- Rewrite the inside inequality as "Q(x)22" and show that it can be reduced to a quadratic inequality A+B+C 0, where the coefficients depend on data and cutoff z but not on p. = (c) The confidence set is unbounded if the coefficient A A(X, 2) is negative. Find the probability2 P(A < 0) as a function of 8. That probability only depends on 6, so plot it over the range 6, E [-3,3]; with z= 1.96. For what values of 6, is that probability relatively high? Give an intuitive explanation for why those 0, values might be problematic.