Suppose Y1 and Y2 are uncorrelated random variables Find a l

Suppose Y1 and Y2 are uncorrelated random variables. Find a linear mean squared error non-homogeneous estimate X_hat in terms of Y1 nad Y2. The estimate has the form X_hat = aY1 + bY2 + c.

Solution

A linear equation in one unknown x may always be rewritten

ax=b

If a 0, there is a unique solution

x=b/a

If a = 0, then, when b = 0 every number is a solution of the equation, but if b 0 there are no solutions.

and a linear function is the form of f(x)=ax+b

and a common form of linear equation is y=mx+b here m,b are parameters.

in general form of linear equation is ax+by+c=0.

using the order of standard form is ax+by=c

in the form of matrix is (A B) (X Y)=C

AX1=BY1=C

we can rewrite the equation and replace X=Y,then

ay1+by2=c

and AY1+BY2+C=0