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626Weighted Centered Sum 1Let X 1, X 2, ... be iid Bernoulli(2/5) random variables, and let F n = sigma(X 1,...,X n). Define M n = sum (k=1) n k*(X k-2/5). Is (M n) a martingale with respect to (F n)?概率简单derivation未尝试免费628Weighted Centered Sum 3Let X 1, X 2, ... be iid Bernoulli(3/7) random variables, and let F n = sigma(X 1,...,X n). Define M n = sum (k=1) n k 2*(X k-3/7). Is (M n) a martingale with respect to (F n)?概率中等derivation未尝试免费630Weighted Centered Sum 5Let X 1, X 2, ... be iid Bernoulli(3/5) random variables, and let F n = sigma(X 1,...,X n). Define M n = sum (k=1) n 3k*(X k-3/5). Is (M n) a martingale with respect to (F n)?概率困难derivation未尝试免费631Terminal-Variable Projection 1Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = 1 X 1+X 2+X 3 >= 2 and M n = E[Y | F n]. Is (M n) a martingale?概率简单derivation未尝试免费632Terminal-Variable Projection 2Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = 1 X 1+X 2+X 3+X 4 = 0 and M n = E[Y | F n]. Is (M n) a martingale?概率中等derivation未尝试免费633Terminal-Variable Projection 3Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = 1 max(X 1,X 2,X 3) = 1 and M n = E[Y | F n]. Is (M n) a martingale?概率中等derivation未尝试免费634Terminal-Variable Projection 4Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = X 1+X 2+X 3+X 4 and M n = E[Y | F n]. Is (M n) a martingale?概率中等derivation未尝试免费635Terminal-Variable Projection 5Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = (X 1+X 2+X 3) 2 and M n = E[Y | F n]. Is (M n) a martingale?概率困难derivation未尝试面试订阅636Normalized Product Process 1Let Y 1, Y 2, ... be iid positive random variables taking values [1, 3] with probabilities ['1/2', '1/2'], and let F n = sigma(Y 1,...,Y n). Define M n = (Y 1...Y n)/(2) n. Is (M n) a martingale?概率简单derivation未尝试免费638Normalized Product Process 3Let Y 1, Y 2, ... be iid positive random variables taking values [1, 4] with probabilities ['3/4', '1/4'], and let F n = sigma(Y 1,...,Y n). Define M n = (Y 1...Y n)/(7/4) n. Is (M n) a martingale?概率中等derivation未尝试免费641Martingale Diagnosis Counterexample 1M n = S n/(n+1), where S n = X 1+...+X n for iid symmetric ±1 increments. Is (M n) a martingale?概率简单derivation未尝试免费642Martingale Diagnosis Counterexample 2M n = X (n+1)-p for iid Bernoulli(p) variables with natural filtration F n. Is (M n) a martingale?概率中等derivation未尝试免费643Martingale Diagnosis Counterexample 3M n = S n 2 - n/2 for a symmetric ±1 random walk. Is (M n) a martingale?概率中等derivation未尝试免费644Martingale Diagnosis Counterexample 4M n = 2 n S n for a symmetric ±1 random walk. Is (M n) a martingale?概率困难derivation未尝试面试订阅646Predictable-Transform Martingale 1M n = sum (k=1) n S (k-1) * X k, where X k are iid symmetric ±1 and S (k-1)=X 1+...+X (k-1). Is (M n) a martingale with respect to the natural filtration?概率简单derivation未尝试免费647Predictable-Transform Martingale 2M n = sum (k=1) n (1+S (k-1) 2) * X k, where X k are iid symmetric ±1. Is (M n) a martingale with respect to the natural filtration?概率简单数值题未尝试免费648Predictable-Transform Martingale 3M n = sum (k=1) n 1 S (k-1)>0 * X k for a symmetric ±1 walk. Is (M n) a martingale with respect to the natural filtration?概率中等derivation未尝试免费649Predictable-Transform Martingale 4M n = sum (k=1) n (2+(-1) k) * X k with iid symmetric ±1 X k. Is (M n) a martingale with respect to the natural filtration?概率中等derivation未尝试免费