Prove, Var(ay)= a^2 var(y) Var(y+a)= var(y) Var(x+y)= var(x)+var(y)+2cov(x,y)
How to prove that: Cov(aX, aY) = a^2Cov(X,Y)
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
For constants a and b, X and Y are random variables. Please prove that, var(aX + bY ) = a 2 var(X) + b 2 var(Y ) + 2abcov(X, Y ) If X and Y are uncorrelated, what will be the results?
CELLANEOUS EXERCISES 1 If X and Y are random variables, prove that brt var
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)
R code please
Problem 3 Use a simulation to verify that when X~N(0,1), Z N(0,1)Y X3+ 10X +Z, we have Var(X+Y) Var(X)+Var(Y)+2Cov(X, Y) and Var(X-Y Var(X)Var(Y)-2Cov(X, Y). (Generate at least 50000 samples.)
6 Suppose that X and Y are random variables such that Var(X) Var(Y)-2 and Cov(x,y)- 1. Find the value of Var(3.X-Y+2)
6 Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(x,y)- 1. Find the value of Var(3.X-Y + 2)
6. Suppose that X and Y are random variables such that Var(X)=Var(y)-2 and Cov(x,y)-1. the value of Var(ax-y-2). Find
Let x be a continuous random variable. Prove var(x) .
(assuming var(x) exist)