Fill in the gaps in the proof of Theorem 6.3.6, Birnbaum's Theorem.
a. Define g(t|θ) = g((j, xj)|θ) = f* ((j, xj)|θ) and

Show that T(j, xj) is a sufficient statistic in the E* experiment by verifying that
g(T(j, xj)|θ)h(j, xj) = g((j, xj)|θ(1) = f* ((j, xj)|θ)
for all (j, xj).
b. As T is sufficient, show that the Formal Sufficiency Principle implies (6.3.4). Also the Conditionality Principle implies (6.3.5), and hence deduce the Formal Likelihood Principle.
c. To prove the converse, first let one experiment be the E* experiment and the other Ej and deduce that Ev(E*, (j, xj)) = Ev(Ej, Xj), the Conditionality Principle. Then, if T(X) is sufficient and T(x) = T(y), show that the likelihoods are proportional and then use the Formal Likelihood Principle to deduce Ev(E, x) = Ev(E, y), the Formal Sufficiency Principle.

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