q4solutions

# q4solutions - Statistics 20 Quiz 4 Solutions November 8...

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Statistics 20: Quiz 4 Solutions November 8, 2006 1. You roll a fair die. For any result X , I will pay you X 2 - 5 dollars. What is your expected winnings for one roll? E ± X 2 - 5 ² = E ± X 2 ² - 5 = h 1 6 (1 2 + 2 2 + 3 3 + 4 2 + 5 2 + 6 2 ) i - 5 = 61 6 . 2. You ﬂip a coin that is weighted to come up heads 75% of the time. You win 5 dollars if it comes up heads and 0 dollars if it comes up tails. What is the variance of your winnings for one ﬂip? Denote your winnings by X and let p = 0 . 75. V ar ( X ) = E [ X 2 ] - ( E [ X ]) 2 = h 1 k =0 (5 k ) 2 P ( X = k ) i - ³ 1 k =0 5 kP ( X = k ) ´ 2 = ± 0 2 (1 - p ) + 5 2 ( p ) ² - (0(1 - p ) + 5( p )) 2 = 25 p - 25 p 2 = 25 p (1 - p ) = 25 * 0 . 75(1 - 0 . 75) = 75 16 = 4 . 6875. 3. A stock broker’s client orders the broker to sell her single share of stock in ﬁve minutes. Dur- ing this time, the stock will either gain 90 dollars with probability 1 / 6, gain 20 dollars with probability 1 / 2, or lose 60 dollars with probability 1 / 3. What is the client’s expected earnings

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q4solutions - Statistics 20 Quiz 4 Solutions November 8...

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