Multivariate Random Variables: formula sheet

Key formulas

Joint PMF (discrete)

f(x, y) = P(X = x, Y = y); f(x, y) ≥ 0; Σx Σy f(x, y) = 1

All cells of the table must add to 1.

Marginal PMF

fX(x) = Σy f(x, y); fY(y) = Σx f(x, y)

Sum across the other variable.

Joint PDF (continuous)

f(x, y) ≥ 0; ∫∫ f(x, y) dx dy = 1 over the whole range

Probabilities come from integrating over a region, not from the value of f.

Marginal PDF

fX(x) = ∫ f(x, y) dy; fY(y) = ∫ f(x, y) dx

Integrate over the full range of the other variable. Watch the limits.

Independence

f(x, y) = fX(x) × fY(y) for all x, y

It must hold for every pair, not just some.

Conditional distribution

f(x | y) = f(x, y) ÷ fY(y), for fY(y) > 0

Links the joint to the marginal.

Conditional probability (mass or density)

f(x | y) = f(x, y) ÷ f(y)

Valid only when f(y) > 0. Each conditional distribution sums (or integrates) to 1.

Marginal from joint

f(x) = Σ f(x, y) over y (discrete); f(x) = ∫ f(x, y) dy (continuous)

Sum across the row or column to get the marginal.

Independence

f(x, y) = f(x) × f(y) for all x, y

Must hold for every pair. One failure means dependent.

Independence, conditional form

f(x | y) = f(x) for all y with f(y) > 0

Equivalent to the product rule.

Conditional expectation (discrete)

E(X | Y = y) = Σ x × f(x | y)

Mean of the conditional distribution.

Law of iterated expectations

E(X) = E[E(X | Y)]

Weight each conditional mean by P(Y = y) and add.

Conditional variance

Var(X | Y = y) = E(X² | Y = y) − [E(X | Y = y)]²

Use conditional probabilities throughout.

Independence and moments

If independent: E(XY) = E(X)E(Y) and Cov(X, Y) = 0

The converse is false in general.

Covariance (definition)

Cov(X,Y) = E[(X − μX)(Y − μY)]

Average product of deviations from the means.

Covariance (shortcut)

Cov(X,Y) = E[XY] − E[X]E[Y]

Fastest form when you have a joint probability table.

Correlation

ρXY = Cov(X,Y) ÷ (σX × σY)

Always between −1 and +1. Needs both standard deviations to be non-zero.

Sample covariance

s_XY = Σ(Xi − X̄)(Yi − Ȳ) ÷ (n − 1)

Divide by n − 1 for the unbiased sample estimate. Divide by n for a population of n equally likely points.

Covariance with itself

Cov(X,X) = Var(X)

Variance is a special case of covariance.

Scaling and shifting

Cov(aX + b, cY + d) = ac × Cov(X,Y)

Constants b and d drop out. Correlation is unchanged if a and c have the same sign and flips sign if they differ.

Variance of a sum

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

For a difference, the covariance term is subtracted: Var(X − Y) = Var(X) + Var(Y) − 2Cov(X,Y).

Independence

X, Y independent ⇒ Cov(X,Y) = 0

The reverse is not true in general.

Mean of a linear combination

E(aX + bY + c) = aE(X) + bE(Y) + c

Always true. No independence needed.

Variance of a weighted sum

Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X, Y)

A constant added to the sum does not change the variance.

Covariance and correlation link

Cov(X, Y) = ρ·σX·σY

Correlation is covariance scaled to lie between −1 and +1.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2

Weights are the portfolio weights. Volatility is σp = √σp².

Variance of a difference

Var(X − Y) = σX² + σY² − 2Cov(X, Y)

The sign of the covariance term flips.

Independent or uncorrelated case

Var(aX + bY) = a²σX² + b²σY²

Valid when Cov(X, Y) = 0. Independence implies zero covariance, but not the reverse.

Matrix form

σp² = wᵀΣw

Σ is the covariance matrix. Diagonal entries are variances.

Skewness

S = E[(X − μ)³] ÷ σ³

Zero for symmetric distributions. Negative means a longer left tail.

Kurtosis

K = E[(X − μ)⁴] ÷ σ⁴

Normal = 3. Always positive.

Excess kurtosis

Excess K = K − 3

Positive means fatter tails than the normal.

Coskewness

S(X,X,Y) = E[(X − μX)²(Y − μY)] ÷ (σX² σY)

Swap the squared variable to get S(X,Y,Y). Powers sum to 3.

Cokurtosis

K(X,X,Y,Y) = E[(X − μX)²(Y − μY)²] ÷ (σX² σY²)

Powers sum to 4. Other versions use powers 3,1 or 1,3.

Sample skewness (equal weights)

Ŝ = [Σ(xi − x̄)³ ÷ n] ÷ σ̂³

Use the same σ̂ convention as the question states.

Covariance from correlation

Cov(X, Y) = ρ × σx × σy

Off-diagonal entry of the covariance matrix. ρ is between −1 and 1.

Conditional mean of Y given X = x

E(Y | X = x) = μy + ρ × (σy ÷ σx) × (x − μx)

Linear in x. The slope ρσy/σx is the regression slope of Y on X.

Conditional variance of Y given X = x

Var(Y | X = x) = σy² × (1 − ρ²)

Does not depend on x. Take the square root for the standard deviation.

Linear combination of two jointly normal variables

aX + bY ~ Normal(aμx + bμy, a²σx² + b²σy² + 2abρσxσy)

Used for portfolio return and risk. Watch the sign of ρ.

Standardisation

Z = (W − mean) ÷ standard deviation

Convert to a standard normal to find probabilities.

Multivariate normal parameters

X ~ N(μ, Σ), where μ is the mean vector and Σ is the covariance matrix

Σ is symmetric. Any linear combination wᵀX is normal with mean wᵀμ and variance wᵀΣw.

Independence rule

For jointly normal X and Y: ρ = 0 ⇔ X and Y are independent

Holds only under joint normality, not for marginally normal variables in general.

Quick revision

• Marginal probability: sum the joint probabilities over the other variable.
• Conditional probability: P(X | Y) = P(X, Y) ÷ P(Y), for P(Y) > 0.
• Independent variables satisfy P(X, Y) = P(X)P(Y) for all values.
• Cov(X, Y) = E(XY) − E(X)E(Y).
• Correlation = Cov(X, Y) ÷ (σX σY), always between −1 and +1.
• Independent variables (with finite variances) have zero covariance, but zero covariance does not imply independence.
• E(aX + bY) = aE(X) + bE(Y), whether or not X and Y are independent.
• Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y).
• Var(X + Y) = Var(X) + Var(Y) if the covariance is zero.
• Covariance changes with units; correlation does not.
• Skewness is the third standardised moment and kurtosis the fourth. A normal has skewness 0 and kurtosis 3.
• For jointly normal variables, zero correlation implies independence.

Common mistakes

• Treating a joint density value f(x, y) as a probability. Fix: For continuous variables, always integrate over a region to get probability.
• Summing the wrong direction when finding a marginal. Fix: To get the marginal of X, you sum over Y, so each X value gets one total.
• Dividing by the wrong marginal, or not dividing at all. Fix: Divide by the marginal of the variable you are conditioning on, the one after the bar.
• Concluding independence from zero correlation. Fix: Zero correlation only rules out linear dependence. Test the product rule. Only for a bivariate normal does zero correlation imply independence.
• Treating zero correlation as independence. Fix: Remember that correlation captures only linear dependence. Y = X² with symmetric X has zero correlation but full dependence.
• Forgetting to subtract E[X]E[Y] when computing covariance. Fix: Always write Cov = E[XY] − E[X]E[Y] first and fill in all three terms.
• Adding standard deviations to get portfolio risk. Fix: Add variances plus the covariance term. Adding σ's works only when ρ = 1 and weights are positive.
• Leaving out the factor 2 on the covariance term. Fix: Always write 2ab·Cov(X, Y) or 2w1w2ρσ1σ2.
• Saying kurtosis of the normal is 0. Fix: Normal kurtosis is 3. Excess kurtosis is 0. Read which one the question asks for.
• Dividing by variance instead of σ³ or σ⁴. Fix: Divide by σ raised to the same power as the moment order.

Exam tips

• Expect a small joint table (2×2 or 3×3) and a question on a marginal, a conditional probability or independence.
• Always write the margins first. Most options are built from common slips such as a joint cell used as a marginal.
• For density questions, check the limits of integration before computing. A triangular support is a common trap.
• If a constant k is unknown, use total probability = 1 before anything else.
• Independence is tested against the product of marginals; zero covariance alone does not prove it.
• Questions often hand you a joint table. Compute marginals first, then the conditional you need.
• For independence, find one cell that fails. It is faster than checking them all.
• Expect a distractor that says uncorrelated variables are independent. It is true only for the bivariate normal.