FRM Exam Part I · Common Univariate Random Variables

Mixture Distributions and Fat Tails for FRM Part I

Updated 11 October 2026 · Fact-checked

A mixture distribution draws each outcome from one of several component distributions, chosen with fixed probabilities. Mixing normals with different variances gives excess kurtosis (fat tails). Mixing normals with different means can add skewness. To solve questions, compute the weighted mean, the weighted second moment, then the variance and kurtosis.

Understand Mixture Distributions and Fat Tails

A mixture distribution is built in two stages. First, a random draw picks a regime, such as calm or stressed, with probabilities w1, w2 and so on. Second, the outcome is drawn from the normal distribution that belongs to that regime. The mixture's density is the probability-weighted average of the component densities.

This is not the same as adding two random variables. A weighted sum of normal variables is still normal. A weighted sum of normal densities is generally not normal. Keep that difference clear.

Fat tails come from mixing different variances. Say 90% of days are calm (low volatility) and 10% are stressed (high volatility). Most outcomes cluster near the centre, but the stressed regime produces extreme moves far more often than a single normal with the same overall variance would. The result has a higher peak and heavier tails. Its kurtosis is above 3, so excess kurtosis is positive. If both components have the same mean, the mixture is symmetric and skewness is zero.

Skewness comes from mixing different means. If the stressed regime has a lower mean and is less likely, the mixture has a long left tail and negative skewness. With equal variances and different means, unequal weights give a skewed mixture, while a 50/50 mix is symmetric. Excess kurtosis in these mixtures can be positive or negative, depending on how far apart the means are and on the weights. For example, a well-separated 50/50 mix of equal-variance normals is bimodal and has kurtosis below 3.

This matters for risk. Financial returns show excess kurtosis and often negative skewness. A mixture of normals is a simple model for that behaviour, because it reflects regime changes. A single normal understates the probability of large losses, so normal-based VaR at high confidence levels can be too low.

Key formulas to remember

Mixture density: f(x) = w1·f1(x) + w2·f2(x), with w1 + w2 = 1; Weights are regime probabilities and must sum to 1. This mixes densities, not random variables.
Mixture mean: μ = w1·μ1 + w2·μ2; The weighted average of component means.
Mixture second moment: E[X²] = w1·(σ1² + μ1²) + w2·(σ2² + μ2²); Use this to get the variance. Do not average the variances alone when means differ.
Mixture variance: σ² = E[X²] − μ²; With equal means this equals w1·σ1² + w2·σ2².
Kurtosis for a zero-mean mixture of normals: Kurtosis = 3·(w1·σ1⁴ + w2·σ2⁴) ÷ (w1·σ1² + w2·σ2²)²; Valid only when both component means are zero. Each normal has fourth moment 3σ⁴. Result is at least 3.
Excess kurtosis: Excess kurtosis = Kurtosis − 3; A normal has kurtosis 3 and excess kurtosis 0.
Skewness: Skewness = E[(X − μ)³] ÷ σ³; Zero when the component means are equal, because the mixture is then symmetric. Different means can make it non-zero.

How to solve Mixture Distributions and Fat Tails questions

Use this method for any question that gives regime probabilities and component parameters.

1Write down the weights and each component's mean and standard deviation. Check the weights sum to 1.
2Compute the mixture mean as the weighted average of the component means.
3Compute the second moment of each component as σ² + μ², then weight them to get E[X²].
4Get the mixture variance as E[X²] − μ². Take the square root if the standard deviation is needed.
5For kurtosis, compute the fourth moment. With zero means, each normal contributes 3σ⁴. Weight them and divide by the variance squared.
6Subtract 3 for excess kurtosis. A value above 0 means fat tails relative to the normal.
7Interpret: equal means give zero skewness; unequal means with a more likely or less likely lower-mean regime give skewness. State the risk implication, such as normal VaR being understated.

Quickest way: Zero-mean shortcut for kurtosis

When to use it: Use when both components have mean zero and different volatilities, which is the typical calm and stressed regime question.

Square each σ to get variances. Compute the weighted variance V = w1·σ1² + w2·σ2².
Square each variance and weight them: W = w1·σ1⁴ + w2·σ2⁴.
Kurtosis = 3·W ÷ V².
If σ1 = σ2, then W = V² and kurtosis is exactly 3. Any difference pushes it above 3.
Eliminate options with kurtosis below 3 or non-zero skewness immediately.

Common mistakes in Mixture Distributions and Fat Tails

Treating a mixture as a weighted sum of random variables, so the result is called normal.
Both use weights, so they look alike.
Fix: A weighted sum of normal variables is normal. A weighted average of densities is a mixture and is not normal unless the components are identical.
Averaging standard deviations instead of variances.
It feels natural to weight the volatilities directly.
Fix: Weight variances (and second moments), then take the square root at the end.
Ignoring the means when computing variance of the mixture.
Students use w1σ1² + w2σ2² for every case.
Fix: Use E[X²] − μ². That shortcut only works when the component means are equal.
Forgetting that each normal has fourth moment 3σ⁴ and using σ⁴.
Kurtosis of 3 for the normal is forgotten inside the formula.
Fix: Multiply by 3 for zero-mean normal components, then divide by variance squared.
Claiming that mixing normals with different variances creates skewness.
Fat tails and skewness are confused.
Fix: With equal means, the mixture is symmetric. Different variances alone give excess kurtosis, not skewness.
Reporting kurtosis when asked for excess kurtosis.
The two terms are used loosely.
Fix: Read the question. Excess kurtosis equals kurtosis minus 3.

Worked examples

Example 1

Daily returns come from a mixture of two zero-mean normals. With probability 80% the standard deviation is 1%. With probability 20% it is 3%. Find the mixture standard deviation and the kurtosis.

Show the solution

Variances: σ1² = 1 (in %²), σ2² = 9.
Mixture variance V = 0.8×1 + 0.2×9 = 0.8 + 1.8 = 2.6. Standard deviation = √2.6 ≈ 1.612%.
Fourth powers: σ1⁴ = 1, σ2⁴ = 81. W = 0.8×1 + 0.2×81 = 0.8 + 16.2 = 17.0.
Kurtosis = 3 × 17.0 ÷ 2.6² = 51 ÷ 6.76 ≈ 7.55.
Excess kurtosis ≈ 7.55 − 3 = 4.55.

Answer: Standard deviation ≈ 1.61%; kurtosis ≈ 7.55 (excess kurtosis ≈ 4.55), with zero skewness.

Example 2

A return is a mixture: with probability 0.75 it is normal with mean 1% and standard deviation 2%; with probability 0.25 it is normal with mean −3% and standard deviation 4%. Find the mixture mean and variance, and find the skewness.

Show the solution

Mean μ = 0.75×1 + 0.25×(−3) = 0.75 − 0.75 = 0%.
Second moments: component 1 = 4 + 1 = 5; component 2 = 16 + 9 = 25.
E[X²] = 0.75×5 + 0.25×25 = 3.75 + 6.25 = 10.
Variance = 10 − 0² = 10 (%²). Standard deviation ≈ 3.16%.
Third central moment: for a normal component with mean deviation d from μ and variance s², E[(X − μ)³] = 3·d·s² + d³. Component 1: d = 1, s² = 4, so 3×1×4 + 1 = 13. Component 2: d = −3, s² = 16, so 3×(−3)×16 − 27 = −171.
E[(X − μ)³] = 0.75×13 + 0.25×(−171) = 9.75 − 42.75 = −33.
Skewness = −33 ÷ 10^1.5 = −33 ÷ 31.62 ≈ −1.04. It is negative.

Answer: Mean 0%, variance 10 (%²), standard deviation ≈ 3.16%; skewness ≈ −1.04 (negative).

Exam tips

If the question gives only different volatilities with equal means, the answer has zero skewness and kurtosis above 3.
Check the means before using the variance shortcut. Unequal means need E[X²] − μ².
Know the link to risk: fat tails mean normal VaR and ES at high confidence levels can be understated.
Keep the arithmetic in %² units, and square-root only at the end. A calculator's memory keys speed this up.
Expect conceptual options too: a mixture of normals is not normal, and it has the same support but heavier tails.

Practice questions from Common Univariate Random Variables

Mixture Distributions and Fat Tails: frequently asked questions

How does a mixture of normals create fat tails?

It combines a high-probability low-volatility regime with a low-probability high-volatility regime. The stressed regime generates extreme outcomes more often than a single normal with the same overall variance. That gives a higher peak and heavier tails, so kurtosis exceeds 3.

Does a mixture distribution always have skewness?

No. If the components share the same mean, the mixture is symmetric and skewness is zero. Skewness needs different means, and the weights and spreads decide its sign.

Is a mixture of two normal distributions itself normal?

Not in general. It is normal only if the two components are identical. Any difference in variance makes kurtosis exceed 3, and a difference in mean can add skewness.

Why does this matter for risk management?

Real returns show excess kurtosis and often negative skewness. A mixture is a simple model of regime changes. Using one normal can understate the chance of large losses and so understate VaR at high confidence levels.