66. Show that the random variables X1,..., Xn are independent if for each i =

2,..., n, Xi is independent of X1,..., Xi−1.

Hint: X1,..., Xn are independent if for any sets A1,..., An P(X j ∈ Aj, j = 1,..., n) = (n j=1 P(X j ∈ Aj)

On the other hand Xi is independent of X1,..., Xi−1 if for any sets A1,..., Ai P(Xi ∈ Ai|X j ∈ Aj, j = 1,...,i − 1) = P(Xi ∈ Ai)