Question

Suppose Microeconomic Principles and Macroeconomic Principles both have two recommended text- books. The bivariate distribution of the number of micro and macro textbooks actually purchased by economics students is: Micro, X Macro, Y X=0 X=1 X = 2 Y = 0 0.15 0.15 0.05 Y =1 0 .15 0.10 0.05 Y = 200.050.30 à. Find the marginal distributions for the number of micro and macro textbooks purchased for each student b. Compute E(X). E(Y), Var(X), Var(Y), and Cov(X,Y). c. Compute Var(X IY). d. Are the munber of micro books purchased and the number of macro books purchased by cach student independent? c. Write down the probability distribution of x conditional on Y=0. 1. Compute E(X Y = 0) and Var(X Y = 0).

Answer 1

[ъge 0 - Given that c910X XOX=1 x=2 I marginal distribution of Y Y=0 / 0.15 0.15 1 0.05 0.35_ ---- Y=1 0.15 0.10 0.05 0.9 0.35 Y=2 0 0.05 10.3 0.30 10.30 10.40 Marginal 1 id distoibution Loft I mongina distoibution of x pqobability 10.3/0.310.4 Marginal distribution of y clo 21 radbabilly 0.35 0.3 0.35 EX EP (X E oxP (x=0)+ 1xP(x+1)+2 xf (x=2) c ox0.3 + 1X0.3 + 9 x0.4 E(X) - 4 E(Y) = Eyp(Y=g) = oxP (Y=0) + 1xP (4=4 )+ 2xP (Y=2)

Page OX0.35 + 1x0.3 + 2 x 0.35 Nospieble (x) = E(X2) - { $(*)] E(x) = & xe (x=<) = oʻxp(Q=0) + 2xP(=1)+23+ (2-2) i 1.9 so, voqiable (x) = E(X2) -(E(x)) - 19- (1.172 = | 599 (t=A)2x3+(1-1)dxz1+0=) a X0 = (h) - voliebre (4) = 107 E(42) - (E(1))2 107-12 = 0.7 cov (x,y) = E(By) - E(-e)Ey} Eley) = (oxo)0.45+ (ixolo.15+ (0x2)0.05+ toxt)0:15 + (1x1) col +(2x1) 0.051 (ox2). + (1x2)0.05+(242) 0.3. = odtool toolt 62 FE(N) = los

Page ① son (x-4) = 15-(1-1)() = 0.4 cor (x,y) = cor (x,y). vogliable (x) vociablecy). 0.4 0.69 X 0.7 c) Vogliable (x+y) = veyable (x) + vesiable (y)+2cc ( = 0.69 +0.70 + 2X0.4 d we know P (oezo, y =o) = 0.15 P (MR20) = 013 , p (9-0)= 0.35 P (x = 0)x p (y = 0) = 0.105 P(x=0.8= 0) = 0.15 + P(:0)*P(4=0) e) P (xly=0) (Y=2, y=0) P(y=0) poobability i P(x=0,Y=0) P(x=1,420) P(40) p(y=0) O

Page ① ons 0.15 0.05 0.35 0.35 0.85 0.43 0.43 10.14 E (XlY=0.) = Erp (x=ze, y=0) 0X0.43t lxo.43 + 2xo.14 = 0.71 vastable (4/4)=o) = E(=0) - {=(xl4-1)}" E (x2/4 =0) = {xe²p (x=xel - 02x0:43 + 12x0.43 + 2²0014 = 0.99,