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FRM Part I · FRM Exam Part I

Random Variables for FRM Part I: Chapter Guide

A random variable assigns a number to each outcome of an uncertain event. For FRM Part I you describe it with a PMF or PDF, a CDF, its mean, variance, skewness and kurtosis, and its relationship to other variables. You solve questions by picking the right formula, substituting carefully and checking the units.

What this chapter covers

This chapter is the base of the Quantitative Analysis topic. It starts with how to describe uncertain outcomes using probability mass functions (discrete variables) and probability density functions (continuous variables). It then moves to the cumulative distribution function, quantiles, expected value, variance and higher moments.

The second half covers two or more variables together: covariance, correlation and joint distributions. It ends with the common distributions you must recognise: uniform, Bernoulli, binomial, Poisson, normal, lognormal, chi-squared, Student's t and F. Each has its own parameters, mean and variance.

The chapter connects to almost everything else in Part I. Value-at-Risk is a quantile of a loss distribution. Portfolio variance uses covariance and correlation. Hypothesis testing and regression rely on the normal, t, chi-squared and F distributions. Market risk and credit risk models in the Valuation and Risk Models topic assume you are comfortable with all of it.

FRM Part I has 100 equally weighted multiple-choice questions in 4 hours, so every question is worth the same and quick, accurate calculation matters. The ideas in this chapter appear directly in Quantitative Analysis questions and indirectly in risk-model and portfolio questions elsewhere. If you master the formulas here, you save time on many other chapters. The questions are usually short and numerical, which makes them reliable marks for a well-prepared candidate.

Random Variables: topics in the order to study them

  1. 1Probability Mass and Density FunctionsStart here because every other idea in the chapter describes or summarises a distribution, and you must know the discrete versus continuous split first.
  2. 2Cumulative Distribution Function and QuantilesThe CDF builds directly on the PMF and PDF, and quantiles are the link to Value-at-Risk later.
  3. 3Expected Value and VarianceOnce you can describe a distribution, you learn its centre and spread, the two most tested summary measures.
  4. 4Moments, Skewness and KurtosisThese extend mean and variance to shape and tail behaviour, so you need the first two moments in place.
  5. 5Covariance, Correlation and Joint DistributionsThis moves from one variable to two, using expected value and variance as the building blocks.
  6. 6Common Probability DistributionsFinish with the named distributions, because you can now apply PMF, CDF, mean, variance and shape to each one.

How to prepare Random Variables

Aim for formula fluency and calculator speed. Most questions are one or two steps, so the goal is to recognise the setup quickly and avoid slips.

  1. Read each topic once for the idea, then write the key formulas from memory: E(X), Var(X) = E(X²) − [E(X)]², Cov(X,Y) and ρ = Cov(X,Y) ÷ (σX × σY).
  2. Practise discrete examples by hand with a small table of outcomes and probabilities. Check that probabilities sum to 1 before you calculate anything.
  3. For continuous variables, practise reading a CDF and finding probabilities as differences, such as P(a < X ≤ b) = F(b) − F(a).
  4. Learn the formulas for skewness and kurtosis, and the benchmark values for the normal distribution: skewness 0 and kurtosis 3 (excess kurtosis 0).
  5. Build a one-page sheet of the common distributions with parameters, mean, variance and typical use. Test yourself until you can fill it in blank.
  6. Use your approved financial calculator for the statistics functions and memory, and practise entering data so it takes seconds, not minutes.
  7. Finish with timed mixed questions. Review every error and note whether it was a concept gap, a formula slip or an arithmetic mistake.

Common mistakes in Random Variables

  • Treating a PDF value as a probability

    Fix: For continuous variables always think of area under the PDF. A single point has probability zero.

  • Forgetting to square the constant in variance

    Fix: Use Var(aX + b) = a²Var(X). The added constant b has no effect on variance.

  • Assuming zero correlation means independence

    Fix: Remember that independence implies zero correlation but not the reverse in general. Nonlinear dependence can leave correlation at zero.

  • Mixing up kurtosis and excess kurtosis

    Fix: Check which one the question states. Excess kurtosis = kurtosis − 3.

  • Dropping the covariance term when adding variances

    Fix: Write Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y) every time, then set covariance to zero only if told to.

  • Confusing variance with standard deviation in answers

    Fix: Underline what the question asks for and check units. Standard deviation is in the same units as the variable; variance is in squared units.

Last-day revision: Random Variables

  • PMF gives P(X = x) for discrete variables; probabilities must be non-negative and sum to 1.
  • PDF gives density, not probability; for a continuous variable P(X = x) = 0 and probability is the area under the curve.
  • CDF: F(x) = P(X ≤ x); it never decreases and runs from 0 to 1.
  • P(a < X ≤ b) = F(b) − F(a).
  • The α-quantile is the value x where F(x) = α; VaR is a quantile of the loss distribution.
  • E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
  • Var(X) = E(X²) − [E(X)]²; standard deviation is √Var(X).
  • Cov(X,Y) = E(XY) − E(X)E(Y); correlation lies between −1 and +1.
  • Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y).
  • Zero correlation does not imply independence; independence does imply zero covariance (where it exists).
  • Normal distribution: skewness 0, kurtosis 3; kurtosis above 3 means fatter tails.
  • Binomial mean = np and variance = np(1 − p); Poisson mean and variance both equal λ.

Random Variables practice questions

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.

Random Variables: frequently asked questions

How much of FRM Part I is Random Variables?

GARP does not publish this chapter as a separate share of the exam, so treat it as foundation material. It sits inside Quantitative Analysis, and its ideas are used in market risk, credit risk and valuation questions as well.

Do I need to memorise the formulas for the common distributions?

Yes, at least the mean and variance of the binomial, Poisson, uniform and normal, and what the t, chi-squared and F distributions are used for. These are short formulas and often decide a question quickly.

Can I use a calculator for these questions?

GARP allows approved financial calculators in the exam. It helps most with means, standard deviations and repeated arithmetic. Practise with your own model so you do not lose time on the day.

What is the best way to understand skewness and kurtosis?

Think of them as shape measures built from the third and fourth moments. Negative skew means a longer left tail, and kurtosis above 3 means fatter tails than the normal. Link both to risk: fat left tails mean larger losses than a normal model predicts.