Bivariate and Multivariate Normal Distributions for FRM Part I

Understand Bivariate Normal and Other Multivariate Distributions

Key formulas to remember

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.

How to solve Bivariate Normal and Other Multivariate Distributions questions

1. 1Write down the given parameters: both means, both standard deviations (not variances) and the correlation. Convert variances to standard deviations if needed.
2. 2Confirm the variables are jointly normal. If the question only says each is normal, you cannot assume the joint properties.
3. 3Identify what is asked: a conditional mean or variance, a portfolio mean or variance, a probability, or a property statement.
4. 4For a conditional question, apply E(Y | X = x) = μy + ρ(σy/σx)(x − μx) and Var = σy²(1 − ρ²).
5. 5For a portfolio or linear combination, compute the mean from the weights, then the variance with the 2abρσxσy term.
6. 6For a probability, standardise with Z = (value − mean) ÷ standard deviation and read the normal table. Use the conditional or the combined distribution, not the original one.
7. 7Check the result: the conditional variance should be no larger than the original, and the portfolio standard deviation should not exceed the weighted sum of standard deviations.

Quickest way: Standardise first, then plug into the formula

Common mistakes in Bivariate Normal and Other Multivariate Distributions

• Using variance instead of standard deviation in the conditional mean formula.

  Questions often give variances, and the formula looks similar to the covariance formula.

  Fix: Take square roots first. The slope is ρ × σy ÷ σx, using standard deviations.

• Forgetting the square on ρ in the conditional variance, writing σy²(1 − ρ).

  The conditional mean uses ρ alone, so students carry that over.

  Fix: Remember it as σy²(1 − ρ²). Sanity check: ρ = 1 should give zero variance.

• Assuming that zero correlation means independence for any two normal variables.

  The rule is true for jointly normal variables and gets over-generalised.

  Fix: State the condition: the variables must be jointly normal. Marginal normality alone is not enough.

• Assuming two normal marginals make a bivariate normal.

  Students treat joint normality as automatic once each variable is normal.

  Fix: Joint normality is a separate assumption. It needs every linear combination to be normal.

• Dropping the covariance term, or using the wrong sign, in portfolio variance.

  Rushing under time pressure, or treating a short position as a positive weight.

  Fix: Always write a²σx² + b²σy² + 2abρσxσy. Use negative weights for short positions and let the sign of ab carry through.

• Believing the normal model captures joint extreme losses.

  The model is so common that its limits are forgotten.

  Fix: Recall that the bivariate normal with |ρ| < 1 has no asymptotic tail dependence. Fat-tailed alternatives such as the multivariate t allow joint extremes.

Worked examples