only do c

1. (Presentation on Riemannian geometry and Fisher information metric) Consider the n-dimensional sphere S" = {(x1, ..., Xn+1) E R"+l; x; + ... + Xn+1 = 1). Take Xi = P; for i = 1, ..., n + 1 with p; > 0. a. What equation does (P1, ..., Pn+1) satisfy? What is the corresponding geometric object? b. Make a parametrization of (the positive part of) S" using (P1, ..., Pn). c. Compute the first fundamental form of S" in this parametrization. This is the Fisher information metric for a discrete random variable with n + 1 values. (A point on the positive part of S" is interpreted as the probability distribution of the random variable.)