a)Cov(X,Y)=Corr(X,Y)*sqrt(Var(X)*Var(Y))=0.25*sqrt(9*4)=1.5
Var(X+Y)=Var(X)+Var(Y)+2*Cov(X,Y) =9+4+2*1.5=16
b)
here E(X(X+Y))=E(X2+XY)=E(X2)+E(XY)=(Var(X)+(E(X))2) +(Cov(X,Y)+E(X)*E(Y))=(9+22)+(1.5+2*0)
=13+1.5=14.5
and E(X)*E(X+Y)=2*(2+0)=4
hence Cov(X,X+Y))=E(X*(X+Y))-E(X)*E(X+Y)=14.5-4=10.5
c)
here Var(X+Y)=Var(X)+Var(Y)+2*Cov(X,Y) =9+4+2*1.5=16
Var(X-Y)=Var(X)+Var(Y)-2*Cov(X,Y) =9+4-2*1.5=10
E((X+Y)*(X-Y))=E(X2-Y2) =(Var(X)+(E(X))2)-(Var(Y)+(E(Y))2) =(9+22)-(4+02)=9
E(X+Y)*E(X-Y)=(2+0)*(2-0)=4
hence Cov((X+Y),(X-Y))=E((X+Y)*(X-Y))-E(X+Y)*E(X-Y)=9-4=5
therefore Corr((X+Y),(X-Y)) =Cov((X+Y),(X-Y))/sqrt(Var(X+Y)*Var(X-Y))
=5/sqrt(16*10)=5/sqrt(160)=0.3953
1. Su the following to 2 decimal places. nE(X) = 2, Var(X) = 9, E(Y) =0,...
X and z are independent. Var(x)=1 ; Var(z) = sigma ^2. E(z)= 0 and y= ax+b+z I) cov(x,y)= ? ii) corr(x,y)=? dependent Varvane 2.
= Var(X) and σ, 1. Let X and Y be random variables, with μx = E(X), μY = E(Y), Var(Y). (1) If a, b, c and d are fixed real numbers, (a) show Cov (aX + b, cY + d) = ac Cov(X, Y). (b) show Corr(aX + b, cY +d) pxy for a > 0 and c> O
4. Assume X ~ Uniform(0, 1) and let Y = 2X+1 and Z = X2 + 1. (a) Find Cov(X,Y), Var(X+Y), Var(X - Y) and Corr(X,Y). (b) Find Cov(X, Z), Var(X + Z), Var(X – Z) and Corr(X, Z).
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.
17. From the following joint probability distribution of X and Y,f (a) P(X= Y], P [X > Y] (b) the distribution of X+ Y (c) E(X), E(Y), Var(X), Var(Y) (d) Cov(X, Y), Corr(X, Y) nd: 0 .3 .25 .05 .2 19 T
(2. Assume that X, Y, and Z are random variables, with EX) = 2, Var(X) = 4, E(Y) = -1, Var(Y) = 6, E(Z) = 4, Var(Z) = 8,Cov(X,Y) = 1, Cov(X, Z) = -1, Cov(Y,Z) = 0 Find E(3X + 4y - 62) and Var(3x + 4y - 62).
For random variables X, Y, and Z, Var(X) = 4, Var(Y) = 9, Var(Z) = 16, E[XY] = 6, E[XZ] = −8, E[Y Z] = 10, E[X] = 1, E[Y ] = 2 and E[Z] = 3. Calculate the followings: (b) Cov(−3Y , −4Z ). (d) Var(Y − 3Z). (e) Var(10X + 5Y − 5Z).
Assume X, Y are independent with EX = 1 EY = 2 Var(X) = 22 Var(Y) = 32 Let U = 2X + Y and V = 2X – Y. (a) Find E(U) and E(V). (b) Find Var(U) and Var(V). (c) Find Cov(U,V).
20. For X let E(X)-0 and sd(x)-2, and for Y let E(Y)--1 and sd(Y)-4. Find: (a) E(X-Y) and E (X Y). (b) Var(X- Y) and Var(X+ Y) if X and Y are independent. (c) EGX+ 흘 Y) and Var(1X+] Y) İf X and Y are independent. (d) Repeat (b) if, instead of independence, Cov(X, Y)- 1. soY is VarY larger
Let(ej denote a white noise process from a normal distribution with E[9] = 0, Var(e-g an Cov(et, e) = 0 for tヂs. Define a new time series {Y.} by Y, = 9 + 0.6 e--04 et-2 + 0.2 9-3 1. Find E(Y) and Var(Y,) 2. Find Cov(Y,X,-k) for k = 1,2,