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FRM Part I · FRM Exam Part I

Multivariate Random Variables for FRM Part I

Multivariate random variables describe two or more uncertain quantities together, such as returns on several assets. To solve questions, read the joint distribution, get marginals and conditionals, then compute covariance, correlation and the mean and variance of a linear combination using Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y).

What this chapter covers

This chapter extends single-variable probability to several variables at once. You start with joint and marginal distributions, then move to conditional distributions and independence. From there you measure how variables move together with covariance and correlation, and use those to find the mean and variance of sums and weighted combinations. The final topics cover higher moments across variables (coskewness and cokurtosis) and the bivariate normal and related multivariate distributions.

Most questions are numerical. You may be given a small joint probability table and asked for a marginal probability, a conditional expectation or a covariance. You may be given volatilities and a correlation and asked for the variance or standard deviation of a two-asset portfolio. Each is a short, mechanical calculation if you know the formula and the setup.

This chapter connects to much of the rest of Part I. Portfolio variance feeds into Foundations of Risk Management, including diversification and the CAPM. Correlation and covariance return in regression, time series and volatility models such as EWMA and GARCH in Quantitative Analysis. Correlation also matters in Valuation and Risk Models, where portfolio VaR and copulas depend on how risks move together. Learn it well here and later chapters become easier.

Part I has 100 equally weighted multiple-choice questions in 4 hours, so every quick, reliable calculation gains marks. This chapter gives you formulas you will use repeatedly, and it supports later topics such as regression, portfolio risk and VaR. GARP publishes no pass mark, so aim for accuracy on the standard calculations rather than guessing which areas matter most. Questions here are usually short and formula-driven, which makes them good value for the study time.

Multivariate Random Variables: topics in the order to study them

  1. 1Joint and Marginal Probability DistributionsEverything else builds on the joint table, and marginals come from summing across it.
  2. 2Conditional Distributions and IndependenceConditionals are joint divided by marginal, so you need the first topic. Independence is then a simple test.
  3. 3Covariance and CorrelationNow you quantify co-movement using expectations taken from the joint distribution.
  4. 4Moments of Sums and Linear Combinations of VariablesThis applies covariance and correlation to portfolio-style variance, the most frequently tested calculation.
  5. 5Skewness, Kurtosis and Coskewness/CokurtosisThese are higher moments. Learn them after the first two moments are secure, and extend them to pairs of variables.
  6. 6Bivariate Normal and Other Multivariate DistributionsIt ties together the earlier ideas in one model and shows where independence and zero correlation do and do not coincide.

How to prepare Multivariate Random Variables

Work from tables to formulas to interpretation. Practise by hand, since the exam gives you a calculator but no spreadsheet.

  1. Build a small joint probability table and practise finding marginals by summing rows and columns. Check that all probabilities add to 1.
  2. Compute conditional probabilities as P(X | Y) = P(X, Y) ÷ P(Y), then test independence by checking whether P(X, Y) = P(X) × P(Y) for every cell.
  3. Calculate covariance from a table using Cov(X, Y) = E(XY) − E(X)E(Y). Then get correlation as Cov(X, Y) ÷ (σX × σY) and check it lies between −1 and +1.
  4. Drill the linear combination formulas until automatic: E(aX + bY) = aE(X) + bE(Y) and Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y). Do many two-asset portfolio examples.
  5. Learn the definitions of skewness and kurtosis (third and fourth standardised moments) and the normal benchmarks: skewness 0, kurtosis 3. Understand what coskewness and cokurtosis measure.
  6. Review the bivariate normal: marginals are normal, and for jointly normal variables zero correlation implies independence. Then do mixed timed practice sets and review every error.

Common mistakes in Multivariate Random Variables

  • Forgetting the covariance term in portfolio variance

    Fix: Always write the full formula with 2ab·Cov(X, Y) first. Drop the term only when you are told the variables are uncorrelated or independent.

  • Dividing by the wrong marginal in a conditional probability

    Fix: Identify what comes after the "given" and use that variable's marginal as the denominator.

  • Claiming zero correlation means independence

    Fix: Remember that correlation captures only linear dependence. In general, independence implies zero correlation, not the reverse.

  • Confusing variance with standard deviation in the final answer

    Fix: Check whether the question asks for variance or volatility, and take the square root only as the last step.

  • Mixing up covariance and correlation

    Fix: Covariance carries units and has no fixed bounds. Correlation is unitless and lies between −1 and +1. Convert using the standard deviations.

  • Mixing up kurtosis and excess kurtosis

    Fix: Check the definition in the question. A normal has kurtosis 3 and excess kurtosis 0.

Last-day revision: Multivariate Random Variables

  • 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.

Multivariate Random Variables practice questions

Multivariate Random Variables in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Multivariate Random Variables: frequently asked questions

How should I solve a joint probability table question?

First check that the probabilities sum to 1. Then get the marginals by summing rows and columns. Use these to find conditionals, expectations and covariance as the question requires.

Why can two variables have zero correlation and still be dependent?

Correlation measures only linear dependence. Two variables can be related in a nonlinear way, such as Y = X² with X symmetric about zero, and still have zero correlation. Independence is a stronger condition.

Do I need to memorise the bivariate normal density?

Focus on its properties rather than the density formula: marginals are normal, conditionals are normal, and zero correlation implies independence. Check the current GARP learning objectives to confirm what is required.

How is this chapter used later in FRM Part I?

Covariance and correlation appear in portfolio risk, regression and volatility models. Joint behaviour of risks also underlies diversification and portfolio VaR in later topics.