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FRM Exam Part I · Random Variables

Cumulative Distribution Function and Quantiles Explained

Updated 11 October 2026 · Fact-checked

The cumulative distribution function (CDF) gives F(x) = P(X ≤ x), the probability that a random variable is at or below x. The inverse CDF reverses this: given a probability p, it returns the quantile x where F(x) = p. To solve, set F(x) equal to p and solve for x.

Understand Cumulative Distribution Function and Quantiles

A random variable can take many values. The cumulative distribution function (CDF) tells you how much probability sits at or below a chosen value. Written F(x) = P(X ≤ x). It works for discrete and continuous variables.

For a discrete variable, you add up the probability mass up to x. The CDF is a step function that jumps at each possible value. For a continuous variable, you integrate the probability density function (PDF) from the lower end up to x. The PDF f(x) is the slope of the CDF. So the PDF gives density at a point, while the CDF gives accumulated probability. For a continuous variable, P(X = x) = 0, and only areas under the PDF are probabilities.

Every CDF is non-decreasing, runs from 0 at the far left to 1 at the far right, and is right-continuous. Because of this, P(a < X ≤ b) = F(b) − F(a). The probability above a value is P(X > x) = 1 − F(x). This is the key to loss tails.

A quantile is the reverse question. Pick a probability p. The p-quantile is the value x such that F(x) = p. The inverse CDF (also called the quantile function) is F⁻¹(p). A percentile is a quantile expressed in percent: the 95th percentile is the 0.95 quantile. The median is the 0.5 quantile.

This matters for risk. Value at Risk is a quantile of the loss distribution. The 99% VaR is the 0.99 quantile of losses, equal to F⁻¹(0.99). If you understand the CDF and its inverse, VaR becomes a simple lookup.

Key formulas to remember

Definition of the CDF
F(x) = P(X ≤ x)
Applies to discrete and continuous variables.
CDF from a PDF (continuous)
F(x) = ∫ f(t) dt, integrated from −∞ to x
For a uniform variable on [a, b], F(x) = (x − a) ÷ (b − a) for a ≤ x ≤ b.
PDF from a CDF
f(x) = dF(x) ÷ dx
Valid where F is differentiable.
CDF from a PMF (discrete)
F(x) = Σ P(X = xᵢ) for all xᵢ ≤ x
A step function that jumps at each possible value.
Interval probability
P(a < X ≤ b) = F(b) − F(a)
For a continuous variable, the endpoints do not matter.
Upper tail
P(X > x) = 1 − F(x)
Used for loss tails and exceedance probabilities.
Quantile (inverse CDF)
x_p = F⁻¹(p), where F(x_p) = p
For a continuous, strictly increasing F the quantile is unique.
Normal quantile
x_p = μ + σ × z_p
z_p = N⁻¹(p). Common values: z(0.95) = 1.645, z(0.975) = 1.96, z(0.99) = 2.326.
Properties of a CDF
0 ≤ F(x) ≤ 1; F(−∞) = 0; F(+∞) = 1; F non-decreasing
A function violating any of these is not a CDF.

How to solve Cumulative Distribution Function and Quantiles questions

Use this routine for any question on CDFs, probabilities and quantiles.

  1. 1Identify whether the variable is discrete or continuous and what is given: a PMF, a PDF, a CDF formula, or a named distribution.
  2. 2Write down what is asked: a probability (value in, probability out) or a quantile (probability in, value out).
  3. 3For a probability, use F(x) for P(X ≤ x), 1 − F(x) for P(X > x), and F(b) − F(a) for P(a < X ≤ b).
  4. 4For a quantile, set F(x) = p and solve for x. For a named distribution, standardise and use the z-value.
  5. 5For a discrete variable, accumulate probabilities in order and pick the smallest x where F(x) reaches or exceeds p.
  6. 6Check the direction of the tail. Loss VaR uses the upper tail, so the 99% quantile has 1% probability above it.
  7. 7Sanity check: the answer must lie in the possible range, and F of your quantile must return p.

Quickest way: Solve F(x) = p directly

When to use it: Use when the CDF is given as a formula or the variable is normal, and the question asks for a quantile or percentile.

  1. Convert the percentile to a decimal p (95th percentile means p = 0.95).
  2. If the CDF is simple (such as uniform), rearrange F(x) = p to get x in one line.
  3. If the variable is normal, use x = μ + σ × z_p with remembered z-values: 1.282 (90%), 1.645 (95%), 1.96 (97.5%), 2.326 (99%).
  4. For left-tail quantiles use the negative z, for example z(0.05) = −1.645.
  5. Plug x back into F to confirm it returns p.

Common mistakes in Cumulative Distribution Function and Quantiles

  • Treating the PDF value as a probability.

    The PDF and CDF are both curves, and f(x) looks like a probability.

    Fix: For a continuous variable only areas under the PDF are probabilities. Use the CDF or integrate. P(X = x) = 0.

  • Using F(x) when the question asks for P(X > x).

    Students rush and read off the left-tail value.

    Fix: Write the event first. P(X > x) = 1 − F(x).

  • Using the wrong tail for a quantile, such as z = −2.326 for 99% loss VaR.

    Confusing return distributions with loss distributions.

    Fix: The 99% quantile of losses has 99% probability below it, so z = +2.326. For returns, the 1% quantile is −2.326 standard deviations from the mean.

  • Forgetting that a discrete CDF is a step function and interpolating.

    Habit from continuous examples.

    Fix: Accumulate the PMF and take the smallest value whose cumulative probability is at least p.

  • Mixing up the CDF and its inverse.

    Both use the letter F and relate x to p.

    Fix: CDF: value in, probability out. Inverse CDF: probability in, value out. Check which one is the input.

  • Saying a CDF can decrease or exceed 1 after wrong algebra.

    Not checking basic properties.

    Fix: Test F(−∞) = 0, F(+∞) = 1, and that F never falls. A candidate function failing this is not a CDF.

Worked examples

Example 1

A loss L is uniformly distributed between $0 and $10 million. Find (a) P(L ≤ 4), (b) P(L > 9), and (c) the 95th percentile of L.

Show the solution
  1. For a uniform variable on [0, 10], F(x) = x ÷ 10 for 0 ≤ x ≤ 10 (units: $ million).
  2. (a) P(L ≤ 4) = F(4) = 4 ÷ 10 = 0.40.
  3. (b) P(L > 9) = 1 − F(9) = 1 − 0.90 = 0.10.
  4. (c) Set F(x) = 0.95. Then x ÷ 10 = 0.95, so x = 9.5.
  5. Check: F(9.5) = 0.95.

Answer: (a) 0.40; (b) 0.10; (c) $9.5 million.

Example 2

Daily portfolio returns are normal with mean 0.05% and standard deviation 1.2%. Find the return level that is exceeded with 95% probability (the 5th percentile of returns), using z(0.95) = 1.645.

Show the solution
  1. The value with 5% probability below it is the 0.05 quantile: x = μ + σ × z(0.05).
  2. By symmetry, z(0.05) = −1.645.
  3. x = 0.05% + 1.2% × (−1.645).
  4. 1.2 × 1.645 = 1.974, so x = 0.05% − 1.974% = −1.924%.
  5. Interpretation: P(return ≤ −1.924%) = 5%, so returns are above −1.924% with 95% probability.

Answer: −1.924%

Exam tips

  • Read whether the question gives a value and wants a probability, or gives a probability and wants a value. This decides CDF versus inverse CDF.
  • Memorise z-values for 90%, 95%, 97.5% and 99%. They save time on every quantile question.
  • Convert tail wording carefully: 'exceeded with 5% probability' and 'at the 95th percentile' describe the same upper quantile.
  • Link to VaR: the VaR at confidence c is the c-quantile of losses, so expect the CDF idea to reappear in risk measure questions.
  • Eliminate options that break CDF properties, such as a value above 1 or a decreasing function.

Practice questions from Random Variables

Cumulative Distribution Function and Quantiles in other exams

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

Cumulative Distribution Function and Quantiles: frequently asked questions

What is the difference between CDF and PDF?

The PDF gives the density of a continuous variable at a point, and probabilities come from areas under it. The CDF gives the accumulated probability up to a value, F(x) = P(X ≤ x). The PDF is the derivative of the CDF, and the CDF is the integral of the PDF.

How do I find a quantile from a CDF?

Set F(x) equal to the target probability p and solve for x. For a normal variable, use x = μ + σ × z_p. For a discrete variable, accumulate probabilities and take the smallest value where the cumulative probability reaches p.

What is the inverse cumulative distribution function?

It is the quantile function F⁻¹(p). You give it a probability and it returns the value x with F(x) = p. For a continuous, strictly increasing CDF it is the exact inverse of F.

Is a percentile the same as a quantile?

They carry the same information. A percentile is a quantile stated in percent, so the 90th percentile is the 0.90 quantile. The median is the 50th percentile.

How does the CDF relate to Value at Risk?

VaR at confidence level c is a quantile of the loss distribution, F⁻¹(c). The 99% VaR is the loss that is exceeded with only 1% probability.