Question

1. Suppose X and Y are discrete random variables with joint probability mass function fxy defined by the following table: 3 y fxy(x, y) 01 3/20 02 10 7/80 3/80 1/5 1/16 3/20 3/16 1/8 2 3 2 3 a Find the marginal probability mass function for X. b Find the marginal probability mass function for Y. c Find E(X), EY],V (X), and V (Y). d Find the covariance between X and Y. e Find the correlation between X and Y. f Are X and Y independent? Justify your answer.

Answer 1

monsider, X and Y are discrete random vanables with joint poobability mass function. 0 1 3/20 2 3 780 stepe 3/80 15 2 - 16 03/16 3/20 78. the marginal distribution of x is. s) The marginal distribution of ï is. Y 0 B 1 2 pcy=y) 19/80 12/80 27 3p1=0) = + 5 cele P14:1) = teme ie hele PLY2) = 4 + + 27 PLY=B) Genealou

© The E(X), ECY), V(x), V(4) _E(X) = 8 x P(X=8)_ A e X 19 + 180 1 X3 + 2 x 33+ 3 x 20 80 80 80 = 144/80 E(x2) = _x2P(X= x)"_ = 0 x 19/80 + 1 x 3780 + 4 X 33/80 + 9x 25/80 CD= 360/80 : : v(x) = E(X%)–(EQ))} = 360 - (none) - 1:26 similary EGY) = J, Y PLNE= )_.. = 0x 19/80 +tX.12780.+ 2x 27/80 + 3 x 22/80_ = 132/80 Ely3 = ply = y). ; . 0 x 19/80.+ 1 x 1780. + 24 27/80 + 9 X 22/80 = 318/80 - v cy) = E(42) - (E())2 = 318 -113212 80 1.80) .1:3193. . : : :

os covariance of 6x.4). - covex, y) = Elxyy - E(X)=E(9). VCOSO Ecxy) - 21 14 p(x=8.Y= Oximo toxo texto xust - 282X6 2x3x + 8x2x28 = 4x5 + 18x4 t 18X5 + 9X10 80 Exiy) = 272... 3 : 80 E(XB) - Ecx). EC4) - cercxuy) = = 272 - 144x132_ - 80 80 972 X80 -144x132 2752 80 X 80: 6400 Covex,y) = 0.43. the covenance between *and V is 0:48: l@ The correlation between x gy is vi coon CX14).s cov (X14)_ - 0:43 vvas(x).VCY V1:26 01:393 = 0.83,250 The correlation between X and Y is 03395 of) The coorelation between X and Y is 0-3335, themefore. x. and Y. ane dypendene, or not. independent