Question 3: Suppose that X and Y are independent random variables, and that their moment generating functions are given by: 1 for t < 1 1 Mx(t) : My(t) = 1 for t < 1- Ot (1— өt)3 Find var(X +Y) Find E(XY) (a) (b)
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- Example 3.6. Let X and Y be jointly continuous random variables with joint PDF is given by: 3 fx.x(x, y) =(1+ a²y)!(0,1)(x)/(0.2)() Note that fx(x) = (1+ x²) I0,1) (2) fy(y) : (1+ y) I(0,2)(y) Solve for fx|y(x | y). Remark. It follows immediately from the definition of conditional probability density function that fx,y(r,y) = fxjy(x | y) · fy (y), fx.x (x, y) = fy|x(y | x) · fx(x), -00(24) Let X be a random variable with p.d.f. 2e-2. * 0A random variable X has a pdf SC(1-x*) -1Let m(t) be the moment generating function of a random variable X. Show that the random variable W = 10X is m(10t). What is the moment generating function of Z = X-5 in terms of m(t)?Ql: () Let X1,X2, X, and X, be four independent random variables, each with pdf f(x)=5(1-x)", 034. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).1. Let X have the moment-generating function Mx (t) = (1– 2t )r/2 , t< ½ (a) Find Var ( -5X) + E(X+2X²) (b) Find the moment-generating function of the random variable Z =X+2Find E(R) and V (R) for a random variable R whose moment-generating function ismR(t) = e2t(1-3t2)-1X is a random variable that has the following PDF function: fx(x) = 0.28(x + 2) + 0.38(x + 1) + 0.358(x) + 0.158(x – 1) %3D For y = x2, find: 1- fy) 2- Fy(y) 3- E[y] 4- oSEE MORE QUESTIONSRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman