Math 408 Actuarial Statistics I A J Hildebrand Variance covariance and moment generating functions Definitions and basic properties Basic definitions Variance Var X E X 2 E X 2 Covariance Cov X Y E XY E X E Y Correlation X Y Cov X Y Var X Var Y Moment generating function mgf M t MX t E etX General properties E c c E cX cE X Var c 0 Var cX c2 Var X Var X c Var X M 0 1 M 0 0 E X M 00 0 E X 2 M 000 0 E X 3 etc E X Y E X E Y Var X Y Var X Var Y 2 Cov X Y Additional properties holding for independent r v s X and Y E XY E X E Y Cov X Y 0 Var X Y Var X Var Y MX Y t MX t MY t Notes Analogous properties hold for three or more random variables e g if X1 Xn are mutually independent then E X1 Xn E X1 E Xn Note that the product formula for mgf s involves the sum of two independent r v s not the product The reason behind this is that the definition of the mgf of X Y is the expectation of et X Y which is equal to the product etX etY In case of indepedence the expectation of that product is the product of the expectations While a variance is always nonnegative covariance and correlation can take negative values 1 Math 408 Actuarial Statistics I A J Hildebrand Practice problems all from past actuarial exams 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 2 The profit for a new product is given by Z 3X Y 5 where X and Y are independent random variables with Var X 1 and Var Y 2 What is the variance of Z 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 X Y Suppose that Var X 5 000 Var Y 10 000 and Var X Y 17 000 If X is increased by a flat amount of 100 and Y is increased by 10 what is the variance of the total benefit after these increases 4 A company insures homes in three cities J K L The losses occurring in these cities are independent The moment generating functions for the loss distributions of the cities are MJ t 1 2t 3 MK t 1 2t 2 5 ML t 1 2t 4 5 Let X represent the combined losses from the three cities Calculate E X 3 5 Given that E X 5 E X 2 27 4 E Y 7 E Y 2 51 4 and Var X Y 8 find Cov X Y X 1 2Y 2
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