Q. 10.15 - A First Course in Probability

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).

Step-by-Step Solution

Answer

(a) The statement is verified and explained below.

(b)The statement is verified and explained below. 

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.