Question

(a) Verify that the minimum of (4.1) occurs when a is as given by (4.2).

(b) Verify that the minimum of (4.1) is given by (4.3).

Short answer

(a) The statement is verified and explained below.

(b)The statement is verified and explained below.

Step-by-step solution

Part (a) Step 1: Given Information

We need to verify the given statement.

Part (a) Step 2: Simplify

From the relation $(4,1)$, we have

$Var\left(W\right)=Var\left[g\left(x\right)\right]+a^{2}Var\left[f9X)\right]+2aCov\left[g\left(X\right),f\left(X\right)\right]$

Consider mapping $a\mapsto Var\left(W\right)$. We are intrested in minimum of this mapping. Using the differentiation, we have that

$\frac{d}{da}Var\left(W\right)=2aVar\left[f\left(X\right)\right]+2Cov\left[g\left(X\right),f\left(X\right)\right]=0$

By solving this equation, it yields that the minimum is obtained at

$a=-\frac{Cov\left[g\left(X\right),f\left(X\right)\right]}{Var\left[f\left(X\right)\right]}$

which have to be proved.

Part (b) Step 1: Given Information

We need to verify the given statement.

Part (b) Step 2: Simplify

To determine the value of the minimum, plug calculated $a$from part (a) into$Var\left(W\right)$. We get

$$\begin{gathered} Var\left(W\right)=Var\left[g\left(X\right)\right]+\frac{Cov{\left[g\left(X\right),f\left(X\right)\right]}^2}{Var{\left[f\left(X\right)\right]}^2}Var\left[f\left(X\right)\right] \\ -2\frac{Cov\left[g\left(X\right),f\left(X\right)\right]}{Var\left[g\left(X\right)\right]}Cov\left[g\left(X\right),f\left(X\right)\right] \\ =Var\left[g\left(X\right)\right]-\frac{Cov{\left[g\left(X\right),f\left(X\right)\right]}^2}{Var\left[f\left(X\right)\right]} \end{gathered}$$

which had to be showed.