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FRM Exam Part I · Common Univariate Random Variables

Normal and Lognormal Distributions for FRM Part I

Updated 11 October 2026 · Fact-checked

A normal variable X ~ N(μ, σ²) is standardized with z = (X − μ) ÷ σ, then read from the standard normal table. A lognormal variable is one whose natural log is normal, so it stays positive. Stock prices are modelled this way. To solve questions, work in logs, then convert back.

Understand Normal and Lognormal Distributions

The normal distribution is the bell-shaped curve defined by two numbers: the mean μ and the standard deviation σ. It is symmetric, with skewness 0 and kurtosis 3. Any linear combination of normal variables is also normal. That is why it is the workhorse of parametric VaR.

To find probabilities you standardize. The z-score z = (X − μ) ÷ σ tells you how many standard deviations X sits from the mean. Z follows the standard normal N(0, 1), and you read probabilities from the table N(z), the cumulative probability. Some values to memorize: roughly 68% of outcomes fall within 1σ of the mean, 95% within 1.96σ, and 99% within 2.576σ.

Confidence intervals and VaR use the same critical values. A two-sided interval is μ ± z × σ. VaR is one-sided, so it uses the one-tail value: 1.645 for 95% and 2.326 for 99%. The 95% two-sided interval uses 1.96 because it leaves 2.5% in each tail. Match the z-value to the tail, not to the confidence percentage alone.

A normal variable can be negative, but a stock price cannot. So we model the log of the price as normal. If ln X is normal, then X is lognormal: always positive and right-skewed. Continuously compounded returns add up over time and are treated as normal, so prices, which are the exponential of those returns, are lognormal.

The key trap is that the mean and variance of a lognormal variable are not simply μ and σ². They are the mean and variance of ln X. The expected price is higher than the median because the right tail is long. The median price is exp(μ), while the mean is exp(μ + σ²/2).

Key formulas to remember

Standardization (z-score)
z = (X − μ) ÷ σ
Z ~ N(0, 1). Then P(X ≤ x) = N(z). For a left-tail probability below the mean, z is negative and N(−z) = 1 − N(z).
Confidence interval for a normal variable
μ ± z × σ
Two-sided critical values: 90% → 1.645, 95% → 1.96, 99% → 2.576.
One-tail critical values (VaR)
95% → 1.645; 97.5% → 1.96; 99% → 2.326; 99.9% → 3.09
Use these for loss at a given confidence level. They are rounded values; exam questions normally give or accept them.
Parametric VaR (normal returns)
VaR = (z × σ − μ) × portfolio value
Often μ is taken as 0 for short horizons. Scale σ by √T for T days if returns are independent.
Linear combination of normals
aX + bY has mean aμx + bμy and variance a²σx² + b²σy² + 2abρσxσy
Holds for jointly normal X and Y. The result is normal.
Lognormal definition
If ln X ~ N(μ, σ²), then X = exp(μ + σZ)
μ and σ are parameters of the log, not of X.
Lognormal mean, median and variance
E[X] = exp(μ + σ²/2); Median = exp(μ); Var(X) = (exp(σ²) − 1) × exp(2μ + σ²)
The mean exceeds the median whenever σ > 0.
Stock price at time T (geometric Brownian motion)
ln(S_T ÷ S_0) ~ N((μ − σ²/2)T, σ²T); E[S_T] = S_0 × exp(μT)
μ is the expected annual return and σ the annual volatility. The median price is S_0 × exp((μ − σ²/2)T).

How to solve Normal and Lognormal Distributions questions

Use this routine for any normal or lognormal question. It keeps the units and the tail straight.

  1. 1Identify the variable that is normal. Is it a return, a P&L, or the log of a price? Write down its mean and standard deviation.
  2. 2Match the horizon. Convert μ by T and σ by √T if the data is annual or daily and the question asks for another period.
  3. 3Decide the tail. Is the question about a loss below a level, a gain above it, or a two-sided interval? Choose the z-value on that basis.
  4. 4Standardize with z = (X − μ) ÷ σ, or go the other way with X = μ + zσ when you are given a probability or confidence level.
  5. 5Read N(z) from the table. For negative z use N(−z) = 1 − N(z). For a right tail use 1 − N(z).
  6. 6For lognormal questions, set the target in logs first: ln(K ÷ S_0). Standardize using the mean (μ − σ²/2)T and standard deviation σ√T.
  7. 7Convert back to the units asked for (dollars, percentage, price) and check the answer is sensible: probabilities between 0 and 1, prices above 0, mean above median for a lognormal.

Quickest way: Memorize the z-table and scale by √T

When to use it: Use it when the question asks for VaR, a confidence interval or a simple tail probability and you have limited time.

  1. Memorize one-tail z-values: 1.28 (90%), 1.645 (95%), 2.326 (99%). Memorize two-sided: 1.96 (95%) and 2.576 (99%).
  2. For VaR, compute z × σ × value, then multiply by √T for a longer horizon.
  3. For a probability, compute z first and compare it with those anchors. If |z| is about 1, the tail is about 16%; if about 2, about 2.3%.
  4. For lognormal means, compute exp(μ + σ²/2) directly. A quick check: the median is exp(μ), and the mean should be slightly larger.
  5. Eliminate options that give the wrong sign, a probability above 1 or the wrong tail before doing the full calculation.

Common mistakes in Normal and Lognormal Distributions

  • Using 1.96 for 95% one-tailed VaR

    1.96 is the most familiar normal value and is tied to 95% in confidence interval questions.

    Fix: VaR is one-sided. At 95% the loss tail is 5%, so use 1.645. Use 1.96 only when 2.5% sits in each tail.

  • Treating the lognormal parameters as the mean and variance of the price

    The letters μ and σ² are reused, so students assume they describe X itself.

    Fix: They describe ln X. The mean of X is exp(μ + σ²/2) and the median is exp(μ). Write 'log-mean' and 'log-variance' next to them in your working.

  • Scaling volatility by T instead of √T

    Mean returns scale with T, so students apply the same rule to the standard deviation.

    Fix: Variance grows with T, so standard deviation grows with √T. A 10-day σ is the 1-day σ × √10, assuming independent returns.

  • Getting the sign wrong in the left tail

    Students look up N(z) for a negative z and the table only lists positive values.

    Fix: Use symmetry: N(−z) = 1 − N(z). Sketch the curve and shade the tail you need before looking anything up.

  • Forgetting the −σ²/2 term in the drift of ln S

    The expected price is S_0 × exp(μT), which has no correction, so students assume the log return has mean μT too.

    Fix: The expected log return is (μ − σ²/2)T. The expected price is exp(μT). Both are right; they answer different questions.

  • Mixing percentage and dollar units in VaR

    Mean and σ are often given in percent while the portfolio is given in dollars.

    Fix: Convert σ to a decimal first (1.2% = 0.012), compute the percentage loss, then multiply by portfolio value.

Worked examples

Example 1

A portfolio's annual return is normally distributed with a mean of 8% and a standard deviation of 15%. What is the probability that the return is below −10%? Options: A) 2.3% B) 8.0% C) 11.5% D) 88.5%

Show the solution
  1. The variable is the annual return, X ~ N(8%, 15%²). We want P(X < −10%).
  2. Standardize: z = (−10 − 8) ÷ 15 = −18 ÷ 15 = −1.2.
  3. From the table, N(1.2) = 0.8849.
  4. By symmetry, N(−1.2) = 1 − 0.8849 = 0.1151.
  5. Check the choices: 88.5% is the right-tail probability for z = +1.2, which is the wrong tail. 2.3% would be z of about −2.

Answer: 11.5% (option C).

Example 2

A stock trades at $100. It follows geometric Brownian motion with expected annual return μ = 10% and volatility σ = 20%. For a one-year horizon, find (a) the expected price, (b) the median price, and (c) the probability that the price ends below $100.

Show the solution
  1. (a) E[S_T] = S_0 × exp(μT) = 100 × exp(0.10) = 100 × 1.10517 = $110.52.
  2. (b) The log return has mean (μ − σ²/2)T = 0.10 − 0.04 ÷ 2 = 0.10 − 0.02 = 0.08, and standard deviation σ√T = 0.20.
  3. The median price is 100 × exp(0.08) = 100 × 1.08329 = $108.33. It is below the mean, as expected for a lognormal variable.
  4. (c) S_T < 100 means ln(S_T ÷ S_0) < ln(1) = 0.
  5. Standardize: z = (0 − 0.08) ÷ 0.20 = −0.4.
  6. N(0.4) = 0.6554, so N(−0.4) = 1 − 0.6554 = 0.3446.

Answer: Expected price $110.52; median price $108.33; probability of ending below $100 is about 34.5%.

Exam tips

  • Write the tail on your scratch paper first (left or right, one- or two-sided). Most wrong answers come from using the wrong z-value or tail.
  • If the question gives you a z-value, use it exactly, even when it differs slightly from your memorized figure.
  • For lognormal items, check whether the question gives the parameters of ln X or of the price. That choice decides whether you add σ²/2.
  • VaR questions often add a horizon scaling step. Apply √T to volatility, not to the VaR formula's z-value.
  • Sense-check each answer: a lognormal mean above the median, a probability below 1, and a VaR that rises with confidence level.

Practice questions from Common Univariate Random Variables

Normal and Lognormal Distributions in other exams

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

Normal and Lognormal Distributions: frequently asked questions

How do I standardize a normal random variable?

Subtract the mean and divide by the standard deviation: z = (X − μ) ÷ σ. The result follows the standard normal distribution with mean 0 and standard deviation 1. You then read the probability from the standard normal table.

What z-values should I memorize for FRM Part I VaR?

The one-tail values are 1.645 for 95% and 2.326 for 99%. For two-sided confidence intervals, use 1.96 for 95% and 2.576 for 99%. Also remember 1.28 for a 90% one-tail value.

What are the mean and variance of a lognormal distribution?

If ln X is normal with mean μ and variance σ², then E[X] = exp(μ + σ²/2) and Var(X) = (exp(σ²) − 1) × exp(2μ + σ²). The median is exp(μ). The mean is larger than the median whenever σ is positive.

Why are stock prices modelled as lognormal?

Continuously compounded returns are treated as normal and add up over time. The price is the exponential of the cumulative return, so it is lognormal. This keeps prices positive and gives a right-skewed distribution, which fits how prices can rise without limit but cannot fall below zero.