Exponential family expectations: Let p(y|φ) = c(φ)h(y) exp{φt(y)} be an exponential family model. a) Take derivatives with respect to φ of both sides of the equation R p(y|φ) dy = 1 to show that E[t(Y )|φ] = −c 0 (φ)/c(φ). b) Let p(φ) ∝ c(φ) n0 e n0t0φ be the prior distribution for φ. Calculate dp(φ)/dφ and, using the fundamental theorem of calculus, discuss what must be true so that E[−c(φ)/c(φ)] = t0.

Given: lines a and b are parallel and line c is a transversal. prove: 2 is supplementary to 8 what is the missing reason in the proof? statement reason 1. a || b, is a transv 1. given 2. ∠6 ≅ ∠2 2. ? 3. m∠6 = m∠2 3. def. of congruent 4. ∠6 is supp. to ∠8 4. def. of linear pair 5. ∠2 is supp. to ∠8 5. congruent supplements theorem corresponding angles theorem alternate interior angles theorem vertical angles theorem alternate exterior angles theorem