Question

4. Two random variables X and Y have the following joint probability density function (PDF) Skx 0<x<y<1, fxy(x, y) = 10 otherwise. (a) [2 points) Determine the constant k. (b) (4 points) Find the marginal PDFs fx(2) and fy(y). Are X and Y independent? (c) [4 points) Find the expected values E[X] and EY). (d) [6 points) Find the variances Var[X] and Var[Y]. (e) [4 points) What is the covariance between X and Y?

Answer 1

0<x<ýsi TUJ I as for valid poly of offer,y) dy de ai 1 [uz. by de a fakelaide ell-x) dle = K[X730*c. oeyes Ek(-3}={K. 36 xx y1 ocacy kale 4 bat ban 342 Ilay) = 6x olxeyal b) f(x) = of Pleylay fox dy 6x [g]' = 6x(1-x) y x = 6x-60 osre fly) = of flery) de Stede b y. 34² ochel for X 84 to be independent -- ----Bar4) flerfly) Xeyare not independent () {(x) = fx fladde & fr. (be - bx) dx = 164 bet 32- 35205 Ely) -- fy-f49dy - 14.3gdy = (3x 3015 a) Elxatz ?. [67-62)de = 16 bar ) 3 ----V(x) - Elx") - LELXl)? 0.5-10.5 -0.05 -------------

E142) Sy? 3ydy Boys 30.5 ---- (4)= EU TELUj? 0.6-10.789% = 0.0375 Les coulx42 Exu) – ELx) ELU) Elxyz @sas s1492.gdyde Sfr y bx dy dx = { box (1) de The Cl- e) de 23}&=xida 2-3(x- 23[***] =3[5- Bayer 0.4 Coulx, 4) = 0.4 - 6.5x0.75 = 0.005