4. Sometimes you eat the bear and sometimes the bear eats you. Suppose that the outcome...

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4. Sometimes you eat the bear and sometimes the bear eats you. Suppose that the outcome of a game is random and 3, X = { with probability 1/2, 1/3, with probability 1/2. a. Calculate E(X) and E [In(X)]. b. Suppose that the game is played n times with each game being independent, and define the payoff to be Yn = X1 X2 x Xn, that is, you bet everything you have each time. Calculate E(Yn) and E [In(Yn)]. ... c. Simulate the game using R and show the distributions of Yn and ln(Yn) after n periods with n = 1, 10, and 100. That is, generate 2,000 different realized values of Yn and ln(Yn), and make a histogram for each n. 4. Sometimes you eat the bear and sometimes the bear eats you. Suppose that the outcome of a game is random and 3, X = { with probability 1/2, 1/3, with probability 1/2. a. Calculate E(X) and E [In(X)]. b. Suppose that the game is played n times with each game being independent, and define the payoff to be Yn = X1 X2 x Xn, that is, you bet everything you have each time. Calculate E(Yn) and E [In(Yn)]. ... c. Simulate the game using R and show the distributions of Yn and ln(Yn) after n periods with n = 1, 10, and 100. That is, generate 2,000 different realized values of Yn and ln(Yn), and make a histogram for each n.