4. You walk into a casino and decide to bet $10 on \'red\' (which has a probabil

Question

4. You walk into a casino and decide to bet $10 on 'red' (which has a probability of 18/37 of coming up) in roulette. If you win, you make a profit of $10 while, if you lose, the casino keeps your $10. Counting a loss as negative, let the random variable Xi be the amount that you gain from the ith bet. You can consider the X, to be independent. (a) What is the expected value of X? (b) What is the variance of X,? (c) Carefully justifying your reasoning, derive the expected total gain from a se- quence of ten bets. from a sequence of ten bets the mean and variance of your total gain compare with those that you calculated (d) Again carefully justifying your reasoning, derive the variance in the total gain (e) Instead of placing ten bets of $10, you decide to place one bet of $100. How do in parts (b) and (c)?

Explanation / Answer

Questiion 4

(a) E[X] = 18/37 * 10 - ( 1- 18/37) * 10 = $ (-10/37)

(b) VaR[X] = 18/37 * (10 + 10/37)2 + 19/37 * (-10 + 10/37) 2 = 99.927

(c) Here expected gain from sequence of 10 bets = -10/37 * 10 = $ (-100/37)

(d) Variace of 10 bets = 10 * 99.927 = 999.277

(e) Here mean bet of $ 100 = 18/37 * 100 - 19/37 * 100 = $ (-100/37)

Variance = 18/37 * (100 + 100/37)2 + 19/37 * (-100 + 100/37) 2 = 9992.695

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