Preview File Edit View Go Tools Window Help 87% Thu 1:36 PM Q E . . . Scannable Document on Oct 22, 2020 at 11_43_00 AM.pdf (page 3 of 5) (2,7 L V Q Search Clear X TH Note: Detailed derivation steps are necessary for credits of a) - d). AM.pdf Suppose X follows a distribution with the density function: Ask Q f ( x| M, 0 2 ) =- V2TO2 e - ( x - 1 ) ? 1 ( 20 2 ). ( * ) Qui a) Show that its moment generating function is Mx (t) =eutter/2. (6 points) (Hint: find the kernel of Normal distribution when you work on the integration) b) Find E(X) and Var(X) with the moment generating function. (6 points) c) Let X and Y be independent random variables that follow density functions fx(x| 0,1) Add and fy(y| 0, 1)respectively as defined in (*). Please note that the functions are both defined in the form of (*). Let Zi=X + Y and Z2= X-Y, are Zi and Z2 independent? (10 points) d) Identify the distribution of Zi and find E(Zi) and Var(Zi). (4 points) (Hint: Only for this subproblem, you can choose the distribution from the four (4) distributions provided in Appendix - I and directly use the given information to answer question d) OCT 22