Quizmaster

Learning objectives

What you will be able to explain

Section 01

Covariance centers the second moment; cross-covariance links two vectors.

Guided checkpoint

Match quantity to definition for random vector $x$ .

Source: Sec. 6.1, pp. 34-35

Uncorrelated does not imply independent without additional distributional assumptions.

Guided checkpoint

Assume finite second moments.

Source: Secs. 6.1-6.2, pp. 34-36

Linearity gives

3(1)-2(-2)+4=11

Guided checkpoint

If $E[x]=[1,-2]^T$ , compute $E[3x_1-2x_2+4]$ .

Source: Sec. 6.2, p. 35

Section 02

Cross terms remain unless uncorrelatedness or independence removes them.

Guided checkpoint

Let $z=Ax+By+c$ .

Source: Sec. 6.2, pp. 35-36

A linear projection turns covariance into a quadratic form.

Guided checkpoint

For scalar $y=a^Tx$ , match each result.

Source: Sec. 6.3, p. 36

a^T Sigma a=2-1-1+3=3

Guided checkpoint

Let $Sigma=[[2,1],[1,3]]$ and $a=[1,-1]^T$ . Compute $Var(a^Tx)$ .

Source: Sec. 6.3, p. 36

Knowledge check

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.