X and z are independent. Var(x)=1 ; Var(z) = sigma ^2. E(z)= 0 and y= ax+b+z...
Prove the following statements
• corr(ax,y) = corr(x,y)
• show that if x,y and z are independent. Show what happened
to:
cov(x+y,x+z)= ?
• assume x and y are not independent:
cov(ax + b, y)= ?
70 tre la Car
1. Su the following to 2 decimal places. nE(X) = 2, Var(X) = 9, E(Y) =0, Var(Y) = 4, and Corr(X,Y) = 0.25. Find a. Var(X +Y) b. Cov(X, X +Y) c. Corr(X+Y,x-Y).
= 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
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.
Let xi be independent. E(xi)=0. Var(xi)= sigma ^2
Cov(x,y) = E(XY) - ExEy
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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).
Obtain E(Z|X), Var(Z|X) and verify that E(E(Z|X)) =E(Z),
Var(E(Z|X))+E(Var(Z|X)) =Var(Z)
3. Let X, Y be independent Exponential (1) random variables. Define 1, if X Y<2 Obtain E (Z|X), Var(ZX) and verify that E(E(Zx)) E(Z), Var(E(Z|X))+E(Var(Z|X)) - Var(Z)
Show that if Y = ax + b (a = 0), then Corr(X, Y = +1 or -1. We know Cov(X, Y) = Covl X, a X ) +o) - (1 ])uxo. X). Then Cov(X, Y) oxor Jux Corr(x, y) = which is 1 when a > 0 and –1 when a < 0 0x (lal ox) lal Under what condition will = +1? The value p = +1 when a > 0
X,Y, and Z are random variables.
Var(X) = 2, Var(Y) = 1, Var(Z) = 5, Cov(X,Y) = 3, Cov(X, Z) = -2, Cov(Y,Z) = 7. Determine Var(3X – 2Y - 2+10)
(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).