Actuarial Statistics · Basic univariate distributions and generating samples

Normal, Lognormal, Exponential and Gamma Distributions

Updated 11 October 2026 · Fact-checked

These are standard continuous models for claim sizes, returns and waiting times. To solve a question, identify the distribution and its parameters, write the pdf or CDF, then use the known mean, variance or moment formula. For the normal and lognormal, standardise and read Φ from the Tables.

Understand Continuous Distributions: Normal, Lognormal, Exponential, Gamma

A continuous distribution describes a variable that can take any value in a range. Probabilities come from areas under the pdf, or directly from the CDF. Each named distribution is a ready-made model with known moments. You do not integrate from scratch unless the question asks you to.

The normal distribution N(μ, σ²) is symmetric and bell-shaped. It suits sums and averages, because of the Central Limit Theorem. You always standardise: Z = (X − μ) ÷ σ. Then you use the standard normal table. The lognormal applies when ln X is normal. It is always positive and right-skewed, so it is used for claim sizes and for investment accumulation factors.

The exponential distribution models the waiting time to one event in a Poisson process. It is memoryless: the time already waited tells you nothing about the time still to wait. The gamma distribution generalises it. If α is a whole number, Gamma(α, λ) is the waiting time to the α-th event, which is the sum of α independent Exp(λ) variables. Exponential is the special case α = 1.

The other distributions are built from these. Chi-square with ν degrees of freedom is a gamma with α = ν/2 and λ = 1/2. It is also a sum of ν squared independent standard normals. The t and F distributions are ratios involving chi-squares, and they are used in inference. The beta lives on (0, 1) and models proportions and probabilities. The Pareto has a heavy right tail, so it suits large losses and reinsurance. The Weibull has a flexible hazard rate and suits lifetimes and claim sizes.

In the Tables, parameters are given in a fixed form. Always check which form is used. For example, the exponential is given with rate λ, not mean. Using the wrong form is the most common source of lost marks.

Key rules to remember

Normal N(μ, σ²): f(x) = [1 ÷ (σ√(2π))] exp(−(x − μ)² ÷ (2σ²)); mean = μ; variance = σ²; MGF = exp(μt + σ²t²/2); The second parameter is the variance, not the standard deviation. Standardise with Z = (X − μ) ÷ σ. Use P(Z < −z) = 1 − Φ(z).
Lognormal (ln X ~ N(μ, σ²)): E[X^k] = exp(kμ + k²σ²/2); mean = exp(μ + σ²/2); variance = exp(2μ + σ²) × (exp(σ²) − 1); median = exp(μ); Get moments from the normal MGF: E[X^k] = E[exp(kY)] with Y ~ N(μ, σ²). Do not use the MGF of X itself.
Exponential(λ): f(x) = λe^(−λx); F(x) = 1 − e^(−λx); S(x) = e^(−λx); mean = 1/λ; variance = 1/λ²; MGF = λ ÷ (λ − t) for t < λ; Memoryless: P(X > s + t | X > s) = P(X > t) = e^(−λt).
Gamma(α, λ): f(x) = λ^α x^(α−1) e^(−λx) ÷ Γ(α); mean = α/λ; variance = α/λ²; MGF = (λ ÷ (λ − t))^α for t < λ; Γ(α) = (α − 1)Γ(α − 1); Γ(n) = (n − 1)! for whole n; Γ(1/2) = √π. A sum of n independent Exp(λ) variables is Gamma(n, λ). Gamma variables with the same λ add: Gamma(α1, λ) + Gamma(α2, λ) = Gamma(α1 + α2, λ) if independent.
Chi-square χ²(ν): Gamma(ν/2, 1/2); mean = ν; variance = 2ν; Z1² + … + Zν² ~ χ²(ν); Used for the distribution of the sample variance: (n − 1)S² ÷ σ² ~ χ²(n − 1) for a normal sample.
t and F distributions: t(ν) = Z ÷ √(χ²(ν) ÷ ν); F(m, n) = [χ²(m) ÷ m] ÷ [χ²(n) ÷ n]; 1 ÷ F(m, n) ~ F(n, m); The t has mean 0 for ν > 1 and variance ν ÷ (ν − 2) for ν > 2. The F has mean n ÷ (n − 2) for n > 2.
Beta(a, b): f(x) = [Γ(a + b) ÷ (Γ(a)Γ(b))] x^(a−1) (1 − x)^(b−1), 0 < x < 1; mean = a ÷ (a + b); variance = ab ÷ ((a + b)²(a + b + 1)); Beta(1, 1) is the uniform distribution on (0, 1).
Pareto(α, λ): f(x) = αλ^α ÷ (λ + x)^(α+1), x > 0; F(x) = 1 − (λ ÷ (λ + x))^α; mean = λ ÷ (α − 1) for α > 1; variance = αλ² ÷ ((α − 1)²(α − 2)) for α > 2; E[X^k] exists only for k < α. It has no MGF, because the tail is too heavy.
Weibull(c, γ): F(x) = 1 − exp(−c x^γ), x > 0; f(x) = cγ x^(γ−1) exp(−c x^γ); mean = c^(−1/γ) Γ(1 + 1/γ); With γ = 1 this is Exponential(c). Check the parameterisation in the Tables before using it.

How to solve Continuous Distributions: Normal, Lognormal, Exponential, Gamma questions

Use this method for any question on a named continuous distribution. It works for probability, moment and parameter-fitting questions.

1Name the distribution from the wording. Look for: sum or average (normal), positive and skewed (lognormal), waiting time (exponential or gamma), heavy tail (Pareto).
2Write down the parameters in the form the Tables use. State clearly whether you have the variance or the standard deviation, and the rate or the mean.
3Decide what is asked: a probability, a percentile, a moment, or a parameter. Choose pdf, CDF, survival function or moment formula to match.
4For the normal, standardise to Z. For the lognormal, take logs of the boundary first, then standardise. Write the z value before reading Φ.
5For moments, use the formula from the Tables. For a lognormal moment use exp(kμ + k²σ²/2). For a gamma, use α/λ and α/λ². Check that the moment exists (for example α > 1 for the Pareto mean).
6For fitting, set the sample mean and variance equal to the formulas and solve for the parameters. Keep the algebra visible.
7Check the answer: probabilities lie in [0, 1], a lognormal mean exceeds its median, and variance is positive. Round only at the end and state the units.

Quickest way: Recognise, convert, read off

When to use it: Use in the multiple-choice section, where each question has about 3 to 4 minutes at most, and for the first line of any written question.

Match the story to the distribution, then write its parameters in one line.
For the normal, compute z = (x − μ) ÷ σ and use symmetry to turn any lower-tail value into a table lookup.
For the lognormal, replace x by ln x, then treat it as a normal question.
For the exponential, skip the integral: P(X > x) = e^(−λx). Use memorylessness to ignore time already elapsed.
For sums, use the relationships: independent normals add means and variances, exponentials with the same rate sum to a gamma, squared standard normals sum to a chi-square.
Reject options that break a basic fact, such as a probability above 1 or a lognormal mean below e^μ.

Common mistakes in Continuous Distributions: Normal, Lognormal, Exponential, Gamma

Treating the second parameter of N(μ, σ²) as the standard deviation, and dividing by 25 instead of 5 (or the reverse).
Many textbooks and software packages use the standard deviation. IAI notation uses the variance.
Fix: Write σ² = value, then σ = √value on the next line, before you standardise.
Giving the lognormal mean as e^μ.
e^μ is the median, and it is easy to confuse the two.
Fix: Use exp(μ + σ²/2) for the mean. Remember the mean is larger than the median because of the right skew.
Using the exponential mean as λ.
Some sources define the exponential by its mean θ. The Tables use the rate λ.
Fix: The mean is 1/λ. If a question says a mean of 5 years, then λ = 0.2 per year.
Applying the Pareto mean or variance formula when α is too small.
Students memorise the formula and ignore the condition on α.
Fix: Check α > 1 for the mean and α > 2 for the variance. If the condition fails, the moment does not exist. Say so.
Forgetting to take logs of the boundary in a lognormal probability.
The question gives x on the original scale, and the student standardises it directly.
Fix: Always convert P(X > x) to P(Y > ln x) first, where Y = ln X is normal.
Mixing up the gamma and Weibull parameterisations, or the degrees of freedom of the chi-square and its gamma parameters.
Several forms of each distribution exist.
Fix: Quote the Tables form at the start. For chi-square, remember α = ν/2 and λ = 1/2.

Worked examples

Example 1

Annual losses on a portfolio, in ₹ thousand, are modelled as N(500, 80²). Find (a) the probability that the loss exceeds 620, and (b) the value x such that the loss is below x with probability 0.95.

Show the solution

Here μ = 500 and σ² = 6,400, so σ = 80.
(a) z = (620 − 500) ÷ 80 = 1.5.
P(X > 620) = 1 − Φ(1.5) = 1 − 0.9332 = 0.0668.
(b) We need Φ(z) = 0.95, so z = 1.6449 from the percentage points table.
x = μ + zσ = 500 + 1.6449 × 80 = 500 + 131.59 = 631.59.

Answer: (a) 0.0668. (b) x ≈ 631.6, that is about ₹6,31,590 if the units are thousands of rupees.

Example 2

A claim size X is lognormal with ln X ~ N(7, 0.25). Find (a) the median, (b) the mean, and (c) P(X > 2,000).

Show the solution

Here μ = 7, σ² = 0.25, so σ = 0.5.
(a) The median is exp(μ) = e^7 = 1,096.6.
(b) The mean is exp(μ + σ²/2) = exp(7 + 0.125) = exp(7.125). Since e^7 = 1,096.63 and e^0.125 = 1.1331, the mean = 1,242.6.
(c) P(X > 2,000) = P(ln X > ln 2,000). ln 2,000 = 7.6009.
z = (7.6009 − 7) ÷ 0.5 = 1.2018, which is about 1.20.
P(Z > 1.20) = 1 − Φ(1.20) = 1 − 0.8849 = 0.1151.

Answer: (a) 1,096.6. (b) 1,242.6, which is above the median as expected. (c) About 0.115.

Exam tips

Write the parameters and their form in the first line of every answer. Examiners award marks for stating the distribution and parameters correctly.
In written questions, derive a moment from the MGF or the integral only if asked to show it. Otherwise quote the Tables result and apply it with clear working.
Lognormal questions often ask for both a probability and a moment. Show ln x, z and Φ(z) on separate lines so that one slip does not cost every mark.
Expect links between distributions, such as a sum of exponentials giving a gamma, or a chi-square from squared normals. Practise proving these using MGFs.
In the computer-based paper, use the R functions dnorm, pnorm, qnorm, plnorm, pexp, pgamma and so on. Note that R's exponential and gamma use the rate by default, and the normal uses the standard deviation.

Practice questions from Basic univariate distributions and generating samples

Continuous Distributions: Normal, Lognormal, Exponential, Gamma in other exams

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

FRM Part INormal and Lognormal Distributions

Continuous Distributions: Normal, Lognormal, Exponential, Gamma: frequently asked questions

How do I find the moments of a lognormal distribution?

Let Y = ln X ~ N(μ, σ²). Then E[X^k] = E[e^(kY)], which is the normal MGF evaluated at t = k. This gives exp(kμ + k²σ²/2). Use k = 1 for the mean and k = 2 for the second moment, then subtract the mean squared for the variance.

What is the difference between exponential and gamma distributions?

The exponential is the waiting time to the first event, with a constant hazard rate. The gamma Gamma(α, λ) is a more flexible shape. For whole α it is the waiting time to the α-th event. The exponential is the gamma with α = 1.

Do I need to memorise the Pareto and Weibull formulae for CS1?

The Formulae and Tables book gives the pdf, CDF and moments, so focus on using them correctly. You must know the parameterisation, the conditions such as α > 1 for the Pareto mean, and the link between the Weibull with γ = 1 and the exponential.

How do I solve normal distribution standardisation problems quickly?

Convert to Z = (X − μ) ÷ σ, sketch the area, and use symmetry so that you only look up positive z values. For percentile questions, find z from the percentage points table and reverse the standardisation: x = μ + zσ.