Question

Let the two-dimensional random variable (X, Y) have the joint density fx.r(x, y) = 16 - x - y)I(0, 2)(x)/(2,4,(y). (a) Find &[Y| X = x]. (6) Find &[Y|X = x]. © Find var (Y|X=x]. (d) Show that &[Y] = { [E[Y|X]]. (e) Find &[XY|X=x]. Tinomial distribution (multinomial with k + 1 =3) of two random variables The trinomial distribution (mu X and Y is given by fx.x(x, y) = x!y!(n - x - y)!' for x, y=0, 1, ..., n and x + x!y!(n-x-vī p*q'(1 - D-7)--» 0, 1, ..., n and x+y Sn, where 0 Sp.0 Sq, and p+asi.

Answer 1

The proof is given in the below attached images.
Given the point pdf of x 8 y os, olg <2 feu) 4 (674) mau marginal pelf of a f(0= f 10. olotot at Noce 1 - 1 1 (141 1831:43] - - [6 (4-2) – 2 (4-2) – ļ (16-4)] [12 - 2x - 2 x 12]. .: f(1) = (3-2) o LXL2 Condition pay of y given x= xu giveness fxy) ${\xx) fo P8-1

5 $(x=). (6-x-4) (3-) - 6-x-4 - 2(3-2) (a). E[//x-x] = { u f(xx-x) dy -S"4(6-709) dy 2 (3-x) s jeu-ay-2). sele 20935". 6046.42.212 28-23736-62-57 1 (52-18x) - 6 (3-2) P82

(26 -9x) = 3(3-X) u (6) E[Y?(x=1] 9 f(y=) dy 24 Ź 2(3-) 2(3-X) Gw, S6Ly93. xL1 LO 2 9.66.150-2003 $u2 - 58x-003 - 213-x) 5213--) { 32.5673 - 3-x) (16 - 28 ) 404- Zx 462-7x)' 3 - 3) 3(3-)

(c) var (N/x-x) = E(Y/x-x] - {E[N/x-x]} 2 4 (12-7x Escás, f4(17-18) - 66-90027 - 3(3-X)/ 363-x) 12/36-33x+722) - (676 +8122_462x)} 3(3-2) = 3(3-X) – (676 + 3(3-) 12-244 +72x ton 9 (3x) 27 - 244 472x + 3x2} (1) 02 - (3 x ² + 72x - 244) = 9 (3-x)2 Marginal of y, a f(x) = f f (x,y) dx -SI (6-x-4) dx $ 6x² (2270 - 4 22 ²2 solo (12-2-24) Pg. 4

FC) = 4(54) E) = Sy sondy sy L (5-4) dy 1 ) ET L 2 3 - + 430 - 563 1 (90-56) 3442-0 [463] [64] BA) da = ut nt al {(26-9x) dx 09262x3² 9. (x)} Es līs P2.5

(52-16) from equo & equo) we get EY] = ELEVAT]. (e.) EN/x-x] = Sxy f/x-x) dy fy. (6-2 9 dy 42 23964-ay-4?) ay - TEI = 60 (52-18x) x (26-90) = 3 (3 - x)