Exercise 4.3 - Correlation coefficient is between -1 and 1

Answers

Without loss of generality, assume $𝔼 [X] = 𝔼 [Y] = 0$ . The statement:

- 1 \leq ρ (X, Y) \leq 1,

is equal to

| ρ (X, Y) | \leq 1 .

Hence we are to prove:

| cov (X, Y) |^{2} \leq var (X) \cdot var (Y)

Which can be drawn straightforwardly from Cauchy–Schwarz inequality. Let

g (t) = t^{2} \cdot var (X) + 2 t \cdot var (X, Y) + var Y = 𝔼 [{(𝑡𝑋 + Y)}^{2}] \geq 0 .

Taking $g$ ’s discriminator finishes the proof.

solour_lfq

2021-03-24 13:42