If the probability distribution of X is given by f(x) = (1/2)x for x = 1, 2, 3, . . . show that E(2X) does not exist. This is the famous Petersburg paradox, according to which a player’s expectation is infinite (does not exist) if he or she is to receive 2x dollars when, in a series of flips of a balanced coin, the first head appears on the xth flip.
Answer to relevant QuestionsWith reference to Definition 4.4, show that µ0 = 1 and that µ1 = 0 for any random variable for which E(X) exists. Definition 4.4 The rth moment about the mean of a random variable X, denoted by µr, is the expected value ...With reference to Exercise 4.8, find the variance of g(X) = 2X + 3. In exercise Use the inequality of Exercise 4.29 to prove Cheby-shev’s theorem. In exercise P(X ≥ a) ≤ µ / a Prove the three parts of Theorem 4.10. Theorem 4.10 If a and b are constants, then 1. 2. 3. Express var(X + Y), var(X – Y), and cov(X + Y, X – Y) in terms of the variances and covariance of X and Y.
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