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].