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(14 pts) (a) (5 pts) Given events A_{1},...,A_{n} in a probability space ? and denote [n] := {1,...,n}. Define A_{I} = ?_{i ? I} A_{i} and B_{I} = ?_{i ? I} A_{i}, where I ? [n] and assume ?_{i ? ?} A_{i} = ? and ?_{i ? ?} A_{i} = ?. We further define the pure slice A°_{I} := A_{I} B_{I'}, where I ? [n], I' = [n] I. Prove that for each I ? [n], P(A_{I}) = ?_{J ? I} P(A°_{J}). In particular, P(?) = ?_{J ? [n]} P(A°_{J}). [Hint: Show that all events A°_{J}, J ? [n], are disjoint, some of them may be empty.] (b) (4=2+2 pts) Let ? = {(i, j) : 1 ? i, j ? 6} be the sample space of rolling a pair of black and white fair dice. Let X, Y be random variables defined by X(i, j) = i and Y(i, j) = j for (i, j) ? ?. (1) Are 3X, 4Y are independent? If yes, verify their independence by definition. (2) Find E((3X + 4Y)²) and V(3X + 4Y). (c) (5 pts) Two gamblers A, B have a, b whole HK dollars respectively, where a, b are positive integers, and bet one dollar each time on a biased HK dollar coin with positive success probability p of the number-side for A and positive success probability q = 1 - p of the flower-side for B. The game continues without stop until A reaches a + b (i.e. B reaches 0) dollars or B reaches a + b (i.e. A reaches 0) dollars. Find the probabilities: (1) A wins all dollars from B. (2) B wins all dollars from A. (3) The game continues forever. (Hint: Let P_{n} denote the probability that A reaches a + b dollars when A is at the situation of holding n dollars, n = 0, 1, 2, ..., a + b.)

          (14 pts) (a) (5 pts) Given events A_{1},...,A_{n} in a probability space ? and denote [n] := {1,...,n}. Define A_{I} = ?_{i ? I} A_{i} and B_{I} = ?_{i ? I} A_{i}, where I ? [n] and assume ?_{i ? ?} A_{i} = ? and ?_{i ? ?} A_{i} = ?. We further define the pure slice A°_{I} := A_{I}  B_{I'}, where I ? [n], I' = [n]  I. Prove that for each I ? [n], P(A_{I}) = ?_{J ? I} P(A°_{J}). In particular, P(?) = ?_{J ? [n]} P(A°_{J}). [Hint: Show that all events A°_{J}, J ? [n], are disjoint, some of them may be empty.]

(b) (4=2+2 pts) Let ? = {(i, j) : 1 ? i, j ? 6} be the sample space of rolling a pair of black and white fair dice. Let X, Y be random variables defined by X(i, j) = i and Y(i, j) = j for (i, j) ? ?. (1) Are 3X, 4Y are independent? If yes, verify their independence by definition. (2) Find E((3X + 4Y)²) and V(3X + 4Y).

(c) (5 pts) Two gamblers A, B have a, b whole HK dollars respectively, where a, b are positive integers, and bet one dollar each time on a biased HK dollar coin with positive success probability p of the number-side for A and positive success probability q = 1 - p of the flower-side for B. The game continues without stop until A reaches a + b (i.e. B reaches 0) dollars or B reaches a + b (i.e. A reaches 0) dollars. Find the probabilities: (1) A wins all dollars from B. (2) B wins all dollars from A. (3) The game continues forever. (Hint: Let P_{n} denote the probability that A reaches a + b dollars when A is at the situation of holding n dollars, n = 0, 1, 2, ..., a + b.)

(14 pts) (a) (5 pts) Given events A1,...,An in a probability space ? and denote [n] := 1,...,n. Define AI = ?i ? I Ai and BI = ?i ? I Ai, where I ? [n] and assume ?i ? ? Ai = ? and ?i ? ? Ai = ?. We further define the pure slice A°I := AI BI', where I ? [n], I' = [n] I. Prove that for each I ? [n], P(AI) = ?J ? I P(A°J). In particular, P(?) = ?J ? [n] P(A°J). [Hint: Show that all events A°J, J ? [n], are disjoint, some of them may be empty.]

(b) (4=2+2 pts) Let ? = (i, j) : 1 ? i, j ? 6 be the sample space of rolling a pair of black and white fair dice. Let X, Y be random variables defined by X(i, j) = i and Y(i, j) = j for (i, j) ? ?. (1) Are 3X, 4Y are independent? If yes, verify their independence by definition. (2) Find E((3X + 4Y)²) and V(3X + 4Y).

(c) (5 pts) Two gamblers A, B have a, b whole HK dollars respectively, where a, b are positive integers, and bet one dollar each time on a biased HK dollar coin with positive success probability p of the number-side for A and positive success probability q = 1 - p of the flower-side for B. The game continues without stop until A reaches a + b (i.e. B reaches 0) dollars or B reaches a + b (i.e. A reaches 0) dollars. Find the probabilities: (1) A wins all dollars from B. (2) B wins all dollars from A. (3) The game continues forever. (Hint: Let Pn denote the probability that A reaches a + b dollars when A is at the situation of holding n dollars, n = 0, 1, 2, ..., a + b.)

Added by Guillermo R.

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