Exercise 4.4 - Correlation coefficient for linearly related variables is 1 or -1

Answers

When $Y = 𝑎𝑋 + b$ :

𝔼 (Y) = a \cdot 𝔼 (x) + b,

var (Y) = a^{2} \cdot var (X) .

Therefore:

\begin{align} 𝑐𝑜𝑣 (X, Y) = & 𝔼 (𝑋𝑌) - 𝔼 (X) \cdot 𝔼 (Y) \\ = & a \cdot 𝔼 (X^{2}) + b \cdot 𝔼 (X) - a \cdot 𝔼^{2} (X) - b \cdot 𝔼 (X) \\ = & a \cdot var (X) . \end{align}

Meanwhile:

var (X) \cdot var (Y) = a^{2} \cdot var (X) .

These two are sufficient to derive:

ρ (X, Y) = \frac{a}{| a |} .

solour_lfq

2021-03-24 13:42