Question

Suppose X, Y are random variables whose joint PDF is given by fxy(x, y) 9 { 0 <y <1,0 < x <y y otherwise 0, 1. Find the covariance of X and Y. 2. Compute Var(X) and Var(Y). 3. Calculate p(X,Y).

Answer 1

sine , we have joint density function fxy OLY <1,0 < ray (2,4)= mel otherwise Note that cov (X,Y) = E(XY) – E(X) ECY) a EXY) - 11 er din diy ST - ľahko ay 1,4 dre dy I dy yzdy I 2 ' 2 [] € (XY) 115 Marginal of y we have joint density OLYLI , oculy Pex,y)= yot XY otherwise

f, y) = To try (34) dr legdut $(") (4-0) = 1 1 fyly): 0x4 1 otherwise ELY) = (484)dy Sy feq) dy = Sydy ។ ] ㅗ 2 Ery) = 100*419) dy = 1) ㅗ 3 vey); pey?) – [€ ex)]* - • -)? *- اما 그 4 11 ㅗ 13 VIY) - 12 (2) Ely) - 2.

How Marginal of t we have joint density 044 < 1 uency for (X,Y)= otherwise o to find independent sange (support ) of * let us plot range of jout density AY ox qal axty yo! (1,1) " LY y = 4 X (1,0) when (Oxual) then fucy 1) ㅠ JyTn) = (feasy, dy ģ ay [dn] [In 1- lineas In(a) 11 -lnke

Ozna! otherwisc = 0+ s fe(a) = S-ln I'm hun die [llen ot á la la E(X)= / - ? 22 dre + re] 2 2 ㅗ Too 4 Since In(l)=0 E 4) = I x? In ze de -- [Amm). 't bo'limi dhe x2 da 11 3 I. 3 [* -o] 1 9 but VIX) = Elx?)-[€(x)] ? 2 =-UP- 5-7 1 16 11 7 144 -- =) ECX) : 7 VCX) 4 144

0 from we have Cov (XY) = EcxY) - E(X). ELY) we have calculated E(X)= ECXY) = 1 = ý ry 2 6 =) CoV (XY) = 16 2 D-2 48 24 we have alredy cale calculated vey) = 1/2 VIX) = 7 144 (3 PCX,Y)= COV (X,Y) s ucx).vcy) 24 = 0.65465 ho 그 12 144 -0.6547