Mean, Variance and Covariance

The core summary statistics of a distribution: variance measures spread, covariance measures co-movement, correlation normalizes it.

1. The idea

Variance measures how spread out a random variable is around its mean: $Var (X) = E [(X - E [X])^{2}] = E [X^{2}] - (E [X])^{2}$ . The second form is usually faster to compute because it avoids centering first.

Covariance extends this to two variables: $Cov (X, Y) = E [X Y] - E [X] E [Y]$ measures whether $X$ and $Y$ tend to move together (positive), oppositely (negative), or show no linear relationship (zero).

Correlation $ρ = Cov (X, Y) / (σ_{X} σ_{Y})$ rescales covariance to lie in $[- 1, 1]$ , making it comparable across variables measured in different units.

2. Why it matters

Variance and covariance are the building blocks of portfolio risk, regression, and every statistical test asked about in research-oriented interviews.

3. Core mathematics

Variance (computational form)

Var (X) = E [X^{2}] - (E [X])^{2}

Covariance

Cov (X, Y) = E [X Y] - E [X] E [Y]

Correlation

ρ_{X, Y} = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}, - 1 \leq ρ \leq 1

Variance of a sum

Var (X + Y) = Var (X) + Var (Y) + 2 Cov (X, Y)

Independence special case

X ⊥ Y ⟹ Cov (X, Y) = 0 ⟹ Var (X + Y) = Var (X) + Var (Y)

4. How to recognize it

Think of this when…

The problem asks for variance, standard deviation, or 'spread' of a quantity
You need Var(X+Y) for dependent variables — covariance term cannot be dropped
Portfolio risk or hedge ratio questions involving two or more assets
You're asked whether zero correlation implies independence (it does not, in general)

5. Worked example

Worked example

Two assets have $Var (X) = 4$ , $Var (Y) = 9$ , and $Cov (X, Y) = - 3$ . Find $Var (X + Y)$ and the correlation $ρ_{X, Y}$ .

1
Apply the variance-of-a-sum formula.
$Var (X + Y) = 4 + 9 + 2 (- 3) = 7$
2
Compute standard deviations: $σ_{X} = 2$ , $σ_{Y} = 3$ .
$σ_{X} = 4 = 2, σ_{Y} = 9 = 3$
3
Divide covariance by the product of standard deviations.
$ρ = \frac{- 3}{2 \cdot 3} = - \frac{1}{2}$
Answer: $Var (X + Y) = 7$ , $ρ_{X, Y} = - 0.5$

6. Common mistakes

Forgetting the $2 Cov (X, Y)$ cross term when computing variance of a sum
Assuming zero covariance implies independence — it only rules out a linear relationship
Using $Var (X - Y) = Var (X) + Var (Y)$ without subtracting the covariance term (it should be minus $2 Cov$ )
Confusing covariance's units (product of the two variables' units) with correlation's unitless scale

7. Quick check

If X and Y are independent with Var(X)=2 and Var(Y)=5, what is Var(X−Y)?

Does Cov(X,Y) = 0 imply X and Y are independent?

8. Practice

24 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Jensen's Inequality

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