The figure is a two dimensional probability simplex

6.D.4^ The purpose of this exercise is to verify the equivalence of the three statements of Proposition 6.D.2 in a two-dimensional probability simplex. Suppose that there are three monetary outcomes: 1, 2, and 3 dollars. Consider the probability simplex in Figure 6.B.1(b). (a) If two lotteries have the same mean, what are their positions relative to each other in the probability simplex. (b) Given a lottery /., determine the region of the simplex in which lie the lotteries L' whose distributions are second-order stochastically dominated by the distribution of L. (c) Given a lottery /., determine the region of the simplex in which lie the lotteries L' whose distributions are mean preserving spreads of L. (d) Given a lottery Z, determine the region of the simplex in which lie the lotteries L' for which condition (6.D.2) holds, where F( . ) and G( ) are, respectively, the distributions of L and 1.. Notice that in (b), (c), and (d), you always have the same region