Consider a probability space with four elements, which we will call a, b, c, and d (i.e. ̩̐ = {a, b, c, d}). The ̩̐-algebra F is the collection of all subsets of ̩̐; i.e., the sets in F are:
̩̐, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a}, {b}, {c}, {d}, ∅.
We define a probability measure P by specifying that P({a}) = 1/6, P({b}) = 1/3, P({c}) = 1/4, P({d}) = 1/4.
We next define two random variables, X and Y, by the following formulas:
X(a) = 1, X(b) = 1, X(c) = -1, X(d) = -1
Y(a) = 1, Y(b) = -1, Y(c) = 1, Y(d) = -1.
We then define Z = X + Y.
List the sets in ̩̐(X).
Determine E[Y|X].
Verify that the partial averaging property is satisfied. That is, show that E[I_A E[Y|X]] = E[I_A Y] for all A ∈ ̩̐(X).