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051Pairwise Independence Check for Dice EventsRoll two fair six-sided dice. Define events A = \ sum is 7\ and B = \ first die shows an even number \ . Are A and B independent? Prove your answer by computing P(A), P(B), and P(A \cap B).概率简单数值题未尝试免费052Pairwise but Not Mutual IndependenceLet the sample space be \Omega = \ 1, 2, 3, 4\ with uniform probability P(\ i\ ) = 1/4. Define A = \ 1, 2\ , B = \ 1, 3\ , C = \ 1, 4\ . Show that A, B, C are pairwise independent but not mutually independent.概率简单derivation未尝试免费053Zero Correlation Does Not Imply IndependenceLet X be uniformly distributed on \ -1, 0, 1\ . Define Y = X 2. Compute Cov (X, Y) and determine whether X and Y are independent.概率中等derivation未尝试免费054Conditional Independence Breaks Under MarginalizationA coin is chosen at random: with probability 1/2 it is fair (p = 1/2) and with probability 1/2 it is biased (p = 1, always heads). Let A be the event that the first flip is heads and B the event that the second flip is heads. Show that A and B are conditionally independent given the coin type C, but that A and B are not marginally independent. Compute P(B \mid A).概率中等derivation未尝试免费055Independent Events Become Dependent Under ConditioningLet A and B be independent events with P(A) = P(B) = 1/2. Define D = A \triangle B (exactly one of A, B occurs). (a) Compute P(D), P(A \mid D), and P(A \cap B \mid D). (b) Are A and B conditionally independent given D? (c) Are A and B conditionally independent given D c?概率困难derivation未尝试面试订阅056Independent Events Cannot Be DisjointSuppose P(A) = 1/3 and P(B) = 1/2. Can A and B be simultaneously independent and mutually exclusive? Prove your answer.概率简单derivation未尝试免费057Independence Is Not TransitiveLet \Omega = \ 1,2,3,4\ with uniform probability. Define A = \ 1,2\ , B = \ 1,3\ , C = \ 3,4\ . (a) Verify that A and B are independent. (b) Verify that B and C are independent. (c) Are A and C independent? What does this say about transitivity of independence?概率中等derivation未尝试免费058XOR Pairwise Independence Without Mutual IndependenceLet X and Y be independent Bernoulli (1/2) random variables. Define Z = X \oplus Y (where \oplus denotes addition modulo 2). (a) Show that Z \sim Bernoulli (1/2). (b) Show that any two of \ X, Y, Z\ are independent. (c) Are X, Y, Z mutually independent? Exhibit a specific joint event that violates the mutual independence condition.概率中等derivation未尝试免费059All Triples Independent but Quadruple NotLet \Omega = \ 0, 1, 2, \ldots, 7\ with uniform probability P(\ \omega\ ) = 1/8. Write each \omega in binary as (b 2, b 1, b 0). Define events: A = \ \omega : b 0 = 1\ , \quad B = \ \omega : b 1 = 1\ , \quad C = \ \omega : b 2 = 1\ , \quad D = \ \omega : b 0 \oplus b 1 \oplus b 2 = 1\ . (a) Show that A, B, C are mutually independent. (b) Show that any triple chosen from \ A,B,C,D\ is mutually independent. (c) Show that A, B, C, D are not mutually independent.概率困难derivation未尝试面试订阅060Independence from Boolean Combinations Requires Mutual IndependenceLet A, B, C be events. (a) Prove that if A, B, C are mutually independent, then A is independent of B \cap C c. (b) Now let \Omega = \ 1,2,3,4\ with uniform probability, A = \ 1,2\ , B = \ 1,3\ , C = \ 1,4\ . Verify that A, B, C are pairwise independent but not mutually independent. (c) Compute P(A \cap (B \cap C c)) and P(A) P(B \cap C c). Does the independence A \perp\!\!\perp (B \cap C c) hold?概率困难derivation未尝试面试订阅061Complements of Independent Events Are IndependentLet A and B be independent events. Prove that A c and B c are also independent, i.e., P(A c \cap B c) = P(A c) P(B c).概率简单derivation未尝试免费062Triple Product Can Hold Without Pairwise IndependenceLet \Omega = \ 1,2,\ldots,8\ with uniform probability. Define events: A = \ 1,2,3,4\ , \quad B = \ 1,2,3,5\ , \quad C = \ 1,4,6,7\ . (a) Show that P(A \cap B \cap C) = P(A)P(B)P(C). (b) Check whether each pair (A,B), (A,C), (B,C) is independent. (c) What does this reveal about the relationship between the triple-product condition and pairwise independence?概率简单derivation未尝试免费063A Function of Independent Variables Need Not Be Independent of Its InputsLet X and Y be independent Bernoulli (1/2) random variables. Define W = \max(X, Y). (a) Compute the distribution of W. (b) Determine whether X and W are independent by checking all four joint probabilities P(X = x, W = w) for x, w \in \ 0, 1\ .概率中等derivation未尝试免费064Independence Is Not Closed Under UnionsLet A, B, C be events with A \perp\!\!\perp B and A \perp\!\!\perp C. (a) Prove that if A, B, C are mutually independent, then A \perp\!\!\perp (B \cup C). (b) Now let \Omega = \ 1,2,3,4\ with uniform probability, A = \ 1,2\ , B = \ 1,3\ , C = \ 1,4\ . Verify that A \perp\!\!\perp B and A \perp\!\!\perp C. (c) Show that A and B \cup C are not independent. Why doesn't pairwise independence suffice?概率中等derivation未尝试免费065Mixtures of Independent Distributions Destroy IndependenceA biased coin lands heads with probability 1/2. If heads, set (X, Y) = (1, 1). If tails, draw X and Y independently, each Bernoulli (1/2). (a) Compute the full joint distribution of (X, Y). (b) Show that X and Y have the same marginal distribution. (c) Are X and Y independent? Prove your answer. (d) Compute P(X = 1 \mid Y = 0) and compare it to P(X = 1). Interpret the result.概率困难derivation未尝试面试订阅066Zero-Probability Events Are Independent of EverythingLet (\Omega, \mathcal F , P) be a probability space and let A be an event with P(A) = 0. (a) Prove that A is independent of every event B \in \mathcal F . (b) Let \Omega = \ 1,2,3,4,5,6\ with uniform probability. Set A = \emptyset and B = \ 1,2,3\ . Verify the independence condition directly. (c) Does the same conclusion hold when P(A) = 1? Prove or give a counterexample.概率简单derivation未尝试免费067Identical Marginals Do Not Imply IndependenceThree cards are labeled (0,0), (0,1), and (1,0). One card is drawn uniformly at random. Let X be the first number and Y the second number on the drawn card. (a) Compute P(X = 0), P(X = 1), P(Y = 0), and P(Y = 1). (b) Are the marginal distributions of X and Y the same? (c) Are X and Y independent? Verify by checking whether P(X = x, Y = y) = P(X = x) P(Y = y) holds for all (x, y) \in \ 0,1\ 2.概率中等derivation未尝试免费068Independence Is Not Hereditary to Sub-EventsLet \Omega = \ 1,2,3,4\ with uniform probability. Define A = \ 1,2\ and B = \ 1,3\ . (a) Verify that A and B are independent. (b) Let A 1 = \ 1\ \subseteq A. Determine whether A 1 and B are independent. (c) State a general principle: if A \perp\!\!\perp B and A 1 \subseteq A, does A 1 \perp\!\!\perp B necessarily hold?概率中等derivation未尝试免费069An Event Independent of Itself Must Be TrivialLet A be an event in a probability space. (a) Write the independence condition P(A \cap A) = P(A) P(A) and deduce which values of P(A) satisfy it. (b) On \Omega = \ 1,2,3,4\ with uniform probability, verify your answer by testing A = \ 1\ , A = \ 1,2\ , A = \emptyset, and A = \Omega. (c) Interpret: what does it mean probabilistically for an event to be independent of itself?概率中等derivation未尝试免费070Pairwise and Triple Independence Without Mutual Independence of Four EventsLet \Omega consist of all binary strings of length 4 with an even number of 1s, each equally likely: \Omega = \ 0000,\, 0011,\, 0101,\, 0110,\, 1001,\, 1010,\, 1100,\, 1111\ . Define events A i = \ \omega \in \Omega : \omega i = 1\ for i = 1,2,3,4. (a) Show that P(A i) = 1/2 for each i. (b) Verify that every pair is independent: P(A i \cap A j) = 1/4 for all i j. (c) Verify that every triple is independent: P(A i \cap A j \cap A k) = 1/8 for all distinct i,j,k. (d) Compute P(A 1 \cap A 2 \cap A 3 \cap A 4) and compare with P(A 1) P(A 2) P(A 3) P(A 4). Are the four events mutually independent?概率困难derivation未尝试面试订阅