1. Let X 0 be uniformly distributed over [0, T], X 1 be uniformly distributed over [0, X 0 ] and, generally, X i+1 be uniformly distributed over [0, X i ], i = 0, 1, ... (1) Prove that the sequence {X...

1. Let X₀
be uniformly distributed over [0, T], X₁
be uniformly distributed over [0, X₀] and, generally, X_i+1
be uniformly distributed over [0, X_i], i = 0, 1, ...

(1) Prove that the sequence {X₀, X₁, ...} is a supermartingale.

2. Let {X₁, X₂, ...} be a homogeneous discrete-time Markov chain with state space Z = {0, 1, ..., n} and transition probabilities

Show that {X₁, X₂, ...} is a martingale. (In genetics, this martingale is known as the Wright-Fisher model without mutation.)