1.*Show that the monotone likelihood ratio condition (f(x)/g(x) increasing in x, where x is a real number) implies first-order stochastic dominance (F(x) G(x) for all x).

2, Let q = q₁ or q2 denote the state of nature in a two state world. Let p₀ = Pr(q = q₁). Let f(p) be an arbitrary distribution on the unit interval such that pf(p)dp = p₀. Show that f(p) can be construed as a distribution over posterior probabilities of p, stemming from an experiment with outcomes y, a prior probability p = p₀ and two likelihood functions p(y|q₁) and p(y|q₂).