Question

(II) Multiple continuous random variables:

8.2 Let X and Y have joint density fXY(x,y) = cx^2y for x and y in the triangle defined by 0 < x < 1, 0 < y < 1, 0 < x + y < 1

and fXY(x,y) = 0 elsewhere.

a. What is c?

b. What are the marginals fX(x) and fY(y)?

c. What are E[X], E[Y], Var[X] and Var[Y]?

d. What is E[XY]? Are X and Y independent?

Answer 1

yCx²dyda - 1x² / ydy dx = 1 Ifnyddyda = 1 (a) f(wy) = Ca?y, ocet, beye si penyayon - Sifacereyan = Sess J'y ay de -4 scf y jdk=1 = $342da 24 ~$ 131,21 = = 2 » [C=6] Hema 419.9)-Gry it (5) f(= 139 dy = fixey dy My = 1 Mangiandofy x+y=1 To Ifa n) = 332 (1-x)? , 02x62 4yly) = 2xy dx = byl ? play 14h, y) = Ży (1-4)!] Ocgel 0 6cm) = föstuda = 4*4*6xly 4* = cyfal.de +64 (C) E(*) = ['xfya) dx = 5'2. 34(1-23 dx 3 / (+13+ 25-2x" ) dx IL - 315 orlu 33t 2(x2) = 5424449dx = 3 5x9+x²–2x) dx '+26-2x dn 2 J f(x) = 0.0285

VLX) = € (m?) - 4€ (X)] = 0.0285 - 60.05) TV (X) = 0.0260 E (Y) = 5'48464) dy = Sy.ayci-yg’dy = 2541-y?- 3y +342)dy = 2 8 (9²-95 _ 3 y² + 3 y") dy [flet "26-% -%]2= EE €9.0 = (603 fip (chEthe-sh-1) 55 J2 = (zh). hp (shet the-gh-sh)} = = 32" = 2[** X 724) *] = 0.0142 NCY] = EC42) - IE cy)] = 0.0142 -(0-0333) ? vey) = 0.0131]

da 2O 4-0 00 431 5 6 7 ky) = $s'a$ 4ony) dy dre - flug. Goly dyda =40***y2 ay) du 26 5 27 ( = -2 57361-3)*dx = 25*241-2232+3x2) da = 2 (83-* - 374 +3855) dx =3 US + {x} = 3 14-4-3+*] | Ecoxy) 30.02141 - As finly) #4 (274(y). So niy are not independent