Q2.

2. Let random variables X, Y, Z, and events {X = x], {Y = y), {Z = z} of positive probability (so that conditional probabilities below are well-defined). Denote the Markov property P[Z = z|Y = y, X = x] = P[Z = z|Y = y] as X - Y - Z. Show that it is equivalent with the following two properties: a) Conditional independence: the future & past are independent given the present (i.e., X, Z are independent given Y): P[X = x, Z = z|Y = y] = P[X = x|Y = y]P[Z = z|Y = y]. b) Reversed Markov property (Z - Y - X), i.e., P[X = x|Y = y,Z = z] = P[X = x|Y = y]