A median of a random variable y or its distribution is any

Question

A median of a random variable Y (or its distribution) is any value m such that P(Y ≤ m) ≥ ½ and P(Y ≥ m) ≥ ½.

(a) Show that the set of medians is a closed interval [m₀, m₁].

(b) Suppose that E|Y| < ∞.="" if="" c="" is="" not="" a="" median="" of="" y,="" show="" that="" e|y="" -="" c|="" ≥="" e|y="" -="" m|="" for="" any="" median="" m="" of="" y.="">

(c) Let X be a sample from P_θ, where θ ∈ Θ ⊂ R. Consider the estimation of θ under the absolute error loss function |a - θ|. Let Π be a given distribution on Θ with finite mean. Find the ℑ-Bayes rule w.r.t. Π, where ℑ is the class of all rules.