Question

4. (30 pts) Let (X,Y) have joint pdf given by e-y, 0 < x < y < 0, f(x,y) = { | 0, 0.w., (a) Find the correlation coefficient px,y. (20 pts) (b) Are X and Y independent? Explain why. (10 pts)

Answer 1

Mean and Variance of Gamma($\alpha, \beta$)

E(2)- fr£)d* Z -1 Z E(22 Z - Z 2 (tds f(arzp)) VE) E(2+)_Ee) : «* ..

े Mong taf१४ : £,)= f)x = ০০ = ८ x~ gpa.(v०l) P=) } E()- 1, Na(x) = | x?০ .. Ma 5 Y: ,७)=$s) ৯, ४ केर = ४० ४ 2-1 ~४ ५ ९ 12 e५, ५० 2. । ४१०

.. [ae*w*]&k [xete]dz ao 3-1 - 2+ 3 d 1 3-1 2 (,x)- E(xy)-E(x)EY)3-121 cor (1 . 2