Practice Problems from Chapters 4 and 5 - Faculty web pages

use of a (transparent) methodologicalmodel in a contract: advantages data and price-fixing mechanismscan be shared between theparties of the contract by using the model, the“haggling” tends to shift awayfrom the price and more on thekey factors that enhances priceaccountability the model gives the opportunityto use the embedded know-howeven to beginners.CustomerModelSupplierSMEF 2005 Roma / 16-17-18 Marzo 2005 6/20

(b) One fuse in part 31a is in a primary system, and the other is in a backup system thatcomes into use only if the primary system fails. The total effective length of life of thetwo fuses is then Y 1 + Y 2 . Find P (Y 1 + Y 2 ≤ 1).32. (WMS7 Exercises 5.79 and 5.93) The random variablesY 1 and Y 2 are uniformly distributed over the triangle atright.y 2(0, 1)(a) Find E(Y 1 Y 2 )(b) Find Cov(Y 1 , Y 2 )(c) Are Y 1 and Y 2 independent?(−1, 0)(1, 0)y 133. (WMS7 Exercise 5.80 and 5.109a) The random variables Y 1 and Y 2 have joint probabilitydensity given byf(y 1 , y 2 ) ={y1 + y 2 if 0 ≤ y 1 ≤ 1 and 0 ≤ y 2 ≤ 10 otherwise.Let T = 30Y 1 + 25Y 2 . Find the expected value and variance of T .34. (WMS7 Exercise 5.82) The random variables Y 1 and Y 2 have joint density function given byf(y 1 , y 2 ) ={1/y1 if 0 ≤ y 2 ≤ y 1 ≤ 10 otherwise.Find E(Y 1 − Y 2 ).35. (WMS7 Exercise 5.95) The discrete random variables Y 1 and Y 2 have the joint probabilityfunctionP (Y 1 = y 1 , Y 2 = y 2 ) = 1 if (y 1 , y 2 ) ∈ {(−1, 0), (0, 1), (1, 0)}.3(a) Find Cov(Y 1 , Y 2 ).(b) Are Y 1 and Y 2 independent?36. (WMS7 Exercise 5.103) Assume Y 1 , Y 2 , and Y 3 are random variables withE(Y 1 ) = 2, E(Y 2 ) = −1, E(Y 3 ) = 4,V (Y 1 ) = 4, V (Y 2 ) = 6, V (Y 3 ) = 8,Cov(Y 1 , Y 2 ) = 1, Cov(Y 1 , Y 3 ) = −1, Cov(Y 2 , Y 3 ) = 0.Find the expected value and variance of 3Y 1 + 4Y 2 − 6Y 3 .