Respuesta :
Answer:
Var(u|X=1) = 1
Var(Y|X=1) = 1
Var(u|X) = 3
Var(Y|X) = 9
Step-by-step explanation:
First, we need to identify the properties of the variance, so:
1. Var(a) = 0 , if a is constant
2. [tex]Var(aX) = a^{2} V(X)[/tex], Where a is constant and X is a random variable
3. [tex]Var(aX+b)=a^{2} Var(X)+0[/tex], Where a and b are constants and X is a random variable.
4. [tex]Var(X + Y)=Var(X) + Var(Y)[/tex], Where X and Y are random variables and are independents.
Then, if Y=1+X+u, where X, Y, and u=v+X are random variables, v is independent of X; E(v)=0, Var(v)=1, E(X)=1, and Var(X)=2, the variance of the following cases are calculated as:
Var(u|X=1) = Var( v + X | X = 1) = Var( v + 1 )
= Var(v) + Var(1)
= 1 + 0 = 1
Var(Y|X=1) = Var( 1 + X + u | X = 1 )
= Var (1 + X + v + X|X=1)
= Var(1 + 1 + v + 1)
= Var(3) + Var(v)
= 0 + 1 = 1
Var(u|X) = Var ( v + X | X)
= Var ( v + X)
= Var(v) + Var(X)
= 1 + 2 = 3
Var(Y|X) = Var ( 1 + X + u | X)
= Var ( 1 + X + v + X | X)
= Var ( 1 + 2X + v)
= Var(1) + Var(2X) + Var(v)
[tex]= Var(1)+2^{2}*Var(X)+Var(v)[/tex]
= 0 + 4*2 + 1 = 8 + 1 = 9
