Skip to content

FRM Part I · FRM Exam Part I

Common Univariate Random Variables for FRM Part I

Common univariate random variables are the standard probability distributions (uniform, Bernoulli, binomial, Poisson, normal, lognormal, chi-squared, t, F and mixtures) used to model one risk factor at a time. To solve questions, identify the distribution, write its parameters, apply the right formula, then compute the probability or moment.

What this chapter covers

This chapter covers the named distributions you will use throughout the FRM. You start with simple discrete cases (Bernoulli, binomial, Poisson), move to continuous ones (uniform, normal, lognormal), and then reach the Central Limit Theorem, the sampling distributions (chi-squared, t, F) and mixture distributions.

Each distribution comes with a short set of facts: what it models, its parameters, its mean and variance, and when it is the right choice. Most questions ask you to recognise the distribution from a story and then do a short calculation.

The chapter connects directly to the rest of Quantitative Analysis. Hypothesis tests and confidence intervals use the normal, t, chi-squared and F distributions. Value at Risk uses the normal and lognormal. Option pricing assumes lognormal prices. Fat tails and mixtures explain why real return data breaks the normal assumption, which matters in market risk and in Valuation and Risk Models.

The exam has 100 multiple-choice questions in 4 hours, so you need fast, reliable recall of formulas. This chapter supplies building blocks for statistics, regression, VaR and option pricing, so a weak grip here costs you marks in several other places. The calculations are short and the formulas are fixed, which makes this one of the most efficient chapters to master. Time spent here pays back across the whole paper.

Common Univariate Random Variables: topics in the order to study them

  1. 1Discrete Uniform and Bernoulli DistributionsThese are the simplest cases and teach the habit of listing outcomes, probabilities, mean and variance.
  2. 2Binomial and Poisson DistributionsBinomial builds on Bernoulli as a sum of independent trials, and Poisson follows as the count-of-events model.
  3. 3Continuous Uniform DistributionIt introduces density and probability as area before you meet the harder continuous shapes.
  4. 4Normal and Lognormal DistributionsThese carry the most exam weight in practice; you need z-scores, standardisation and the link between normal log returns and lognormal prices.
  5. 5Central Limit TheoremIt explains why the normal appears so often, and it needs the normal distribution first.
  6. 6Chi-Squared, Student's t and F DistributionsThese are built from normal variables and sample statistics, so they come after the normal and the CLT.
  7. 7Mixture Distributions and Fat TailsMixtures combine distributions you already know and show where the normal fails, so they close the chapter.

How to prepare Common Univariate Random Variables

Aim to know each distribution well enough to name it from a short description and compute the answer in about a minute or two.

  1. Build a one-page sheet listing each distribution with its use, parameters, mean and variance.
  2. For each distribution, solve three or four numerical problems by hand, writing the formula, the numbers and the answer.
  3. Practise normal calculations until standardising, z = (x − μ) ÷ σ, and reading a z-table are automatic.
  4. Learn the link between lognormal prices and normal log returns, and practise moving between the two.
  5. For the CLT and sampling distributions, focus on conditions, the standard error σ ÷ √n and how degrees of freedom change the shape.
  6. Review fat tails and mixtures conceptually: know what excess kurtosis means and why mixtures create heavier tails.
  7. Finish with timed mixed sets so you practise recognising the distribution before calculating.

Common mistakes in Common Univariate Random Variables

  • Mixing up the binomial and Poisson models.

    Fix: Use binomial when there is a fixed number of trials n with probability p. Use Poisson when events occur at an average rate λ over an interval with no fixed n.

  • Forgetting to standardise, or using variance instead of standard deviation in z.

    Fix: Check whether the question gives σ or σ². Take the square root if needed before computing z.

  • Treating lognormal parameters as the mean and volatility of the price.

    Fix: State clearly that the parameters describe ln(X), and remember that the mean of X is higher than e^μ.

  • Applying the CLT to a single observation or to very small samples.

    Fix: Remember it concerns the distribution of the sample mean for large n, with the standard error σ ÷ √n.

  • Using the wrong degrees of freedom for t and chi-squared.

    Fix: For a single sample mean or variance, use n − 1. Check the setup each time.

  • Assuming fat tails mean a larger variance.

    Fix: Fat tails mean more extreme outcomes relative to the normal, shown by kurtosis above 3, even when the variance is the same.

Last-day revision: Common Univariate Random Variables

  • Bernoulli: P(1) = p, mean p, variance p(1 − p).
  • Binomial: mean np, variance np(1 − p); n independent trials with the same p.
  • Poisson: mean and variance both equal λ; models counts of events in a fixed interval.
  • Discrete uniform on n equally likely values has probability 1 ÷ n for each.
  • Continuous uniform on [a, b]: mean (a + b) ÷ 2, variance (b − a)² ÷ 12.
  • Normal: fully described by mean and variance; skewness 0 and kurtosis 3.
  • Standardise with z = (x − μ) ÷ σ before using normal tables.
  • If log returns are normal, the price is lognormal and cannot be negative.
  • CLT: the sample mean of independent, identically distributed observations with finite variance is approximately normal for large n, with standard error σ ÷ √n.
  • Student's t has heavier tails than the normal and approaches it as degrees of freedom grow.
  • Chi-squared is a sum of squared standard normals and is used for variance tests; F is a ratio of two scaled chi-squared variables.
  • Mixtures of normals with different variances give fat tails and excess kurtosis.

Common Univariate Random Variables practice questions

Common Univariate 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.

Common Univariate Random Variables: frequently asked questions

How many distributions do I need to memorise for FRM Part I?

You need the main ones in this chapter: Bernoulli, binomial, Poisson, uniform, normal, lognormal, chi-squared, t and F, plus the idea of mixtures. For each, know the use, parameters, mean and variance. A one-page sheet is enough to organise this.

Do I need to use a table or calculator for normal probabilities?

Know the common z-values for the normal distribution, such as those used for 95% and 99% confidence. Use a financial calculator for arithmetic like powers, exponentials, logarithms and square roots. Practise these before exam day.

Why does the lognormal distribution matter in FRM?

Asset prices cannot go below zero, and the lognormal respects that. When continuously compounded returns are normal, prices are lognormal. This is the base for option pricing and many risk models.

How does this chapter connect to hypothesis testing?

Test statistics follow the normal, t, chi-squared or F distributions. If you know where each comes from and how degrees of freedom work, the testing chapters become much easier.