I would appreciate having a solution written with nice handwriting and detail explanations

2. Let Q be a sample space on Which a probability has been dened. (Assume that Q is a nite set if you want.) Let A be an interval and 7' : Q X A > R be a function. For every a E A, dene Ta : Q + R by i310) : r(w,a) for every (.0 E 9. (Usually people use \"7'(-, a)\" to refer to Ta.) Assume that for every 0, E A, Ta is a random variable and it has an expectation. Also, assume that for all m E 9, (1,6 E A and t 6 [0,1], 1(a), 156 + (1 t)a) 2 t'r'(w, b) + (1 t)?'(w, a). (1) (In other words, T(w, -) is a concave function on A for every w E 9.) Dene R : A > R as R(a) : Efra] for every (1 E A. (a) Show that R is a concave function. (h) Assume that Eq. (1) holds with strictly inequality whenever 0. 7E b and t 6 (0,1). Can you deduce that R is strictly concave