Q1) Consider the space of two coin tosses Ω2 = {HH , HT, TH, TT} and let stock prices be given by S0 = 4, S1 (H) = 8, S1(T) = 2, S2(HH) = 16, S2(HT) = S2(TH) = 4, S2(TT) = 1.
Consider two probability measures given by
P1(HH) = 1/4, P1(HT) = 1/4, P1(TH) = 1/4, P1(TT) = 1/4,
P2 (HH) = 4/9, P2(HT) = 2/9, P2(TH) = 2/9, P2(TT) = 1/9,
Define the random variable X = 1 if S2 = 4 otherwise 0.
(i) List all the sets in σ(X).
(ii) List all the sets in σ (S1).
(iii) Show that σ(X) and σ (S1) are independent under the probability measure P1.
(iv) Show that σ(X) and σ(S1) are not independent under the probability measure P2.
(v) Under P2 we have P2{S1= 8} = 2/3 and P2{S1 =2} = 1/3. Explain intuitively why, if you are told that X = 1, you would want to revise your estimate of the distribution S1.
Q2) Consider a probability measure Ω with four elements, which we call a, b, c, and d. The σ-field F is the collection of all subsets of Ω.
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,
And as usual, the probability of other sets in F is the sum of the probabilities of the elements in the set. We next define two random variables X and Y by the 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
(i) List all the sets in σ(X).
(ii) Determine E[Y|X], specify the values of this random variable for a, b, c, and d. Verify that the partial-averaging property is satisfied.
(iii) Determine E [Z|X]. Verify the partial-averaging property.
(iv) Compute E[Z|X] - E[Y|X], Using the property of conditional expectations, explain why you get X.
Q3) Let X and Y be random variables on the probability space (Ω, F, P). Then Y = Y1+ Y2, where Y1= E[Y|X] is σ(X)-measurable and Y2= Y - E[Y|X] . Show that Y2 and X are uncorrelated. More generally, show that Y2 is uncorrelated with every σ(X)-measurable random variable.
Q4) Let (Ω, F, P) be Uniform measure on [0; 1) with the Borel σ-algebra. Suppose also that X (ω) = ω3/4 and that G = σ ([0, 1/3), [1/3, 2/3), [2/3, 1)): Explicitly find E[X|G].