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 – and Z = 1 if the two tosses yielded an identical result; otherwise, it is 0.

1 1) Describe 1 = σ(X1) and 2 = σ(X2). Is X1 2-measurable?

2 2) Describe = σ(Y). Is Y 1-measurable? Is X1 -measurable?

3 3) Describe = σ(Z). Is Z 1-measurable, -measurable? Is X1 -measurable?

4 4) Give the inclusions between , 1, 2, and .

nn≥1

nn∈ℕnn∈ℕnn∈ℕ

EXERCISE 1.7.– Consider the following game of chance. A player begins by choosing a number between 6 and 8 (inclusive), which we call the principal. The player then rolls 2 uncut, six-sided, non-rigged dice and sums the result. The wins are as follows:

 – If the sum is 2 or 3, the player loses 1 DT (Tunisian dinar).

 – If the sum is 11, the player wins 1 DT if the principal is 7; otherwise, they lose 1 DT.

 – If the sum is 12, the player wins 1 DT if the principal is 6 or 8; otherwise, they lose 1 DT.

 – Finally, in all other cases, nothing happens (no win, no loss).

1 1) Determine Ω, the universe of all outcomes of the experiment.

2 2) S is the random variable giving the sum of the two dice. Determine the distribution of S.