Previous QuizLecture 7 Examples\fbox{\parbox{1\textwidth}{**Questions have a number in '()' bracket. This is the question number of the sample set of questions for Exam P found online on the SOA website.}}Math 370X - Lecture 7 ExamplesJoint, Marginal and Conditional DistributionsSaumil PadhyaOctober 24, 2016Previous Quiz4.The lifetime of a machine part is exponentially distributed with a mean of five years. Calculatethe mean lifetime of the part, given that it survives less than ten years.Hint:f(X|X < a) =f(x)P (X < a)E[X|X < a] =aZ−∞x · f(x|x < a) dx(A) 0.865(B) 1.157(C) 2.568(D) 2.970(E) 3.4355.A certain town experiences an average of 5 tornadoes in any four year period. The numberof years from now until the town experiences its next tornado as well as the number of yearsbetween tornados have identical exponential distributions and all such times are mutuallyindependent Calculate the median number of years from now until the town experiences itsnext tornado.(A) 0.55(B) 0.73(C) 0.80(D) 0.87(E) 1.251Lecture 7 Examples**Questions have a number in ’()’ bracket. This is the question number of the sample set ofquestions for Exam P found online on the SOA website.Example 1:(108) A device containing two key components fails when, and only when, bothcomponents fail. The lifetimes, X and Y of these components are independent with joint densityfunctionf(x, y) = Ke−xe−yfor x > 0, y > 0The cost, Z, of operating the device until failure is 2X+Y. Let g be the density function for Z.Determine g(z), for z > 0.(A) e−z/2− e−z(B) 2(e−z/2− e−z)(C)z2e−z2(D)e−z/22(E)e−z/33Example 2:(230) Let X denote the proportion of employees at a large firm who will choose tobe covered under the firm’s medical plan, and let Y denote the proportion who will choose to becovered under both the firm’s medical and dental plans. Suppose that for 0≤ y ≤ x ≤1, X andY have the joint cumulative distribution functionF (x, y) = y(x2+ xy − y2)Calculate the expected proportion of employees who will choose to be covered under both plans.(A) 0.06(B) 0.33(C) 0.42(D) 0.50(E) 0.752Example 3: (118) Let X and Y be continuous random variables with joint density functionf(x, y) = 15y, x2≤ y ≤ xLet g be the marginal density function of Y.Determine which of the following represents g.(A) g(y) = 15y, 0 < y < 1(B) g(y) =15y22, x2< y < x(C) g(y) =15y22, 0 < y < 1(D) g(y) = 15y3/2(1 − y1/2), x2< y < x(E) g(y) = 15y3/2(1 − y1/2), 0 < y < 13Example 4: (110) Let X and Y be continuous random variables with joint density functionf(x, y) = 24xy, 0 < x < 1, 0 < y < 1 − xCalculate P[Y<X|X=1/3].(A) 1/27(B) 2/27(C) 1/4(D) 1/3(E) 4/9Example 5:(114) A diagnostic test for the presence of a disease has two possible outcomes: 1for disease present and 0 for disease not present. Let X denote the disease state (0 or 1) of apatient, and let Y denote the outcome of the diagnostic test. The joint probability function of Xand Y is given by:P [X = 0, Y = 0] = 0.800P [X = 1, Y = 0] = 0.050P [X = 0, Y = 1] = 0.025P [X = 1, Y = 1] = 0.125Calculate Var(Y|X=1).(A) 0.13(B) 0.15(C) 0.20(D) 0.51(E) 0.71Example 6:(106) Let X and Y denote the values of two stocks at the end of a five-year period.X is uniformly distributed on the interval (0, 12). Given X = x, Y is uniformly distributed on theinterval (0, x).Calculate Cov(X, Y) according to this model.(A) 0(B) 4(C) 6(D) 12(E) 244Example 7:(165) Two claimants place calls simultaneously to an insurer’s claims call center.The times X and Y, in minutes, that elapse before the respective claimants get to speak withcall center representatives are independently and identically distributed. The moment generatingfunction of each random variable isM(t) =11 − 1.5t2, t <23Calculate the standard deviation of X + Y.(A) 2.1(B) 3.0(C) 4.5(D) 6.7(E)
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