Math 408, Actuarial Statistics I A.J. HildebrandVariance, covariance, and moment-generating functionsDefinitions and basic properties• Basic definitions:– Variance: Var(X) = E(X2) − E(X)2– Covariance: Cov(X, Y ) = E(XY ) − E(X)E(Y )– Correlation: ρ = ρ(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) = c2Var(X), Var(X + c) = Var(X)– M (0) = 1, M0(0) = E(X), M00(0) = E(X2), M000(0) = E(X3), 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, . . . , Xnare 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 + Yis the expectation of et(X+Y ), which is equal to the product etX· etY. In case ofindepedence, the expectation of that product is the product of the expectations.– While a variance is always nonnegative, covariance and correlation can take negativevalues.1Math 408, Actuarial Statistics I A.J. HildebrandPractice pr obl ems (all from past actuarial exams)1. Suppose that the cost of maintaining a car is given by a random variable, X, with mean200 and variance 260. If a tax of 20% is introducted on all items associated with themaintenance 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 independentrandom 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 thatVar(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 varianceof the total benefit after these increases?4. A company insures homes in three cities, J, K, L. The losses o cc urring in these cities areindependent. The moment-generating functions for the loss distributions of the citiesareMJ(t) = (1 − 2t)−3, MK(t) = (1 − 2t)−2.5, ML(t) = (1 − 2t)−4.5Let X represent the combined losses from the three cities. Calculate E(X3).5. Given that E(X) = 5, E(X2) = 27.4, E(Y ) = 7, E(Y2) = 51.4 and Var(X + Y ) = 8,find Cov(X + Y, X + 1.2Y
View Full Document
Unlocking...