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Unformatted text preview: Math 408, Actuarial Statistics I A.J. Hildebrand Variance, covariance, and moment-generating functions Practice problems Solutions 1. Suppose that the cost of maintaining a car is given by a random variable, X , with mean 200 and variance 260. If a tax of 20% is introducted on all items associated with the maintenance of the car, what will the variance of the cost of maintaining a car be? Solution: The new cost is 1 . 2 X , so its variance is Var(1 . 2 X ) = 1 . 2 2 Var( X ) = 1 . 44 260 = 374. 2. The profit for a new product is given by Z = 3 X- Y- 5, where X and Y are independent random variables with Var( X ) = 1 and Var( Y ) = 2. What is the variance of Z ? Solution: Using the properties of a variance, and independence, we get Var( Z ) = Var(3 X- Y- 5) = Var(3 X- Y ) = Var(3 X )+Var(- Y ) = 9Var( X )+Var( Y ) = 11 . 3. An insurance policy pays a total medical benefit consisting of a part paid to the surgeon, X , and a part paid to the hospital, Y , so that the total benefit is...
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