CFA Level I Exam · Statistical Distributions for Financial Asset Prices and Returns

Discrete and Continuous Uniform Distributions for CFA Level I

Updated 7 October 2026 · Fact-checked

A uniform distribution gives every outcome in its range the same chance. For a discrete uniform with n outcomes, each has probability 1/n. For a continuous uniform on [a, b], P(x1 ≤ X ≤ x2) = (x2 − x1) ÷ (b − a), mean = (a + b) ÷ 2, and variance = (b − a)² ÷ 12.

Understand Discrete and Continuous Uniform Distributions

A uniform distribution is the simplest probability model. Every outcome, or every equal-width slice of the range, is equally likely. Nothing is more probable than anything else.

A discrete uniform distribution has a finite list of outcomes, each with the same probability. A fair six-sided die is the standard case: each face has probability 1/6. If there are n outcomes, each has probability 1/n.

A continuous uniform distribution spreads probability evenly over an interval from a to b. Its density is a flat line at height 1/(b − a). Because the total area under the density must equal 1, the height is fixed by the width. For a continuous variable, the probability of any single exact value is 0. Only intervals have probability.

This makes probability a matter of proportion. The chance that X falls in an interval equals the interval's length divided by the total length. If the range is 0 to 10 and you want the chance of landing between 2 and 5, that is 3 ÷ 10 = 0.30.

Uniform distributions matter for the exam because they are easy to test with a calculator and because random number generators in Monte Carlo simulation draw from a continuous uniform distribution on [0, 1]. Those draws are then transformed into other distributions.

Key formulas to remember

Discrete uniform probability: P(X = xi) = 1 ÷ n; n is the number of equally likely outcomes. Cumulative probability for k outcomes up to and including a value is k ÷ n.
Discrete uniform mean (consecutive integers 1 to n): E(X) = (n + 1) ÷ 2; Applies only when outcomes are the integers 1, 2, ..., n. For other lists, compute the simple average of the outcomes.
Discrete uniform variance (consecutive integers 1 to n): Var(X) = (n² − 1) ÷ 12; Same condition as above. Standard deviation is the square root.
Continuous uniform density: f(x) = 1 ÷ (b − a) for a ≤ x ≤ b, and 0 otherwise; The height is constant across the interval.
Continuous uniform probability: P(x1 ≤ X ≤ x2) = (x2 − x1) ÷ (b − a); Requires a ≤ x1 ≤ x2 ≤ b. If an interval extends outside [a, b], cut it back to the range first.
Continuous uniform CDF: F(x) = (x − a) ÷ (b − a) for a ≤ x ≤ b; F(x) = 0 below a and 1 above b.
Continuous uniform mean: E(X) = (a + b) ÷ 2; The midpoint of the range.
Continuous uniform variance: Var(X) = (b − a)² ÷ 12; Standard deviation = (b − a) ÷ √12.

How to solve Discrete and Continuous Uniform Distributions questions

Use this method for any uniform distribution question. The key is to decide first whether the variable is discrete or continuous.

1Decide the type. A list of separate outcomes (die, numbered items) is discrete. A measurement over a range (time, return between limits) is continuous.
2Write down the parameters: n for discrete, or a and b for continuous.
3Identify what is asked: a probability, a mean, a variance, or a standard deviation.
4For a probability, count favourable outcomes ÷ n (discrete) or compute interval length ÷ (b − a) (continuous).
5For a continuous interval, clip it to [a, b] if it runs outside the range.
6For mean and variance, use (a + b) ÷ 2 and (b − a)² ÷ 12 for continuous. For discrete, use the formulas for 1 to n, or average the listed values directly.
7Check that the answer is between 0 and 1 for a probability, and that variance is not confused with standard deviation.
8Choose the option that matches, then eliminate the other two using common traps such as forgetting to divide by 12.

Quickest way: Length-ratio shortcut

When to use it: Use for any continuous uniform probability or mean and variance question when time is short.

Compute the range width: b − a.
For probability, divide the width of the target interval by b − a.
For mean, take the midpoint of a and b.
For variance, square the width, then divide by 12. For standard deviation, divide the width by √12 ≈ 3.464.
Reject any option that falls outside 0 to 1 for a probability, or that equals the width squared with no division by 12.

Common mistakes in Discrete and Continuous Uniform Distributions

Forgetting to divide by 12 in the variance, or using (b − a)² ÷ 2.
Students remember the squared width but mix this up with other formulas.
Fix: Memorise Var = (b − a)² ÷ 12 as one unit. Test it on [0, 1]: variance should be 1/12 ≈ 0.0833.
Reporting variance when the question asks for standard deviation.
Students stop one step early under time pressure.
Fix: Underline the word asked. Standard deviation = (b − a) ÷ √12.
Not clipping the interval to the range.
A question asks for P(X > 8) when the range is 2 to 10, and students use the wrong upper limit.
Fix: Replace limits outside [a, b] with a or b, then use length ÷ (b − a).
Using the 1-to-n discrete formulas on outcomes that are not consecutive integers from 1.
The formula (n + 1) ÷ 2 looks general.
Fix: For other lists such as 10, 20, 30, 40, average the values directly and compute variance from deviations.
Giving a nonzero probability for one exact value of a continuous variable.
Students carry over the discrete idea of 1/n.
Fix: For a continuous variable, P(X = x) = 0. Only intervals have positive probability.

Worked examples

Example 1

The time a trade takes to settle is uniformly distributed between 2 and 10 minutes. What is the probability that a trade settles in more than 7 minutes?

Show the solution

Identify the type: continuous uniform with a = 2 and b = 10.
Range width = 10 − 2 = 8.
The event X > 7 corresponds to the interval from 7 to 10, width = 3.
Probability = 3 ÷ 8 = 0.375.

Answer: 0.375 (37.5%)

Example 2

A continuous random variable is uniformly distributed between 4 and 16. What are its mean and standard deviation?

Show the solution

Parameters: a = 4, b = 16.
Mean = (4 + 16) ÷ 2 = 10.
Width = 16 − 4 = 12.
Variance = 12² ÷ 12 = 144 ÷ 12 = 12.
Standard deviation = √12 ≈ 3.46.

Answer: Mean = 10; standard deviation ≈ 3.46 (variance = 12)

Exam tips

Numerical options are listed from smallest to largest, so once you have computed a value you can locate it quickly. The order does not tell you whether an option is correct, so always rely on your own calculation.
A common wrong option is the variance when the standard deviation is asked, or the width squared without the 12. Check which one you computed.
For discrete questions with a fair die or numbered items, count favourable outcomes and divide by n. This takes under 30 seconds.
Remember that a continuous uniform on [0, 1] is the basis of Monte Carlo random draws. A conceptual question may test this link.

Practice questions from Statistical Distributions for Financial Asset Prices and Returns

Discrete and Continuous Uniform Distributions in other exams

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

Discrete and Continuous Uniform Distributions: frequently asked questions

What is the difference between discrete and continuous uniform distributions?

A discrete uniform has a finite set of outcomes, each with probability 1/n. A continuous uniform spreads probability evenly over an interval, so probability depends on interval length. Single exact values have probability 0 in the continuous case.

How do I calculate probability for a continuous uniform distribution?

Divide the length of the target interval by the total length b − a. Clip the interval to the range [a, b] if it extends outside. For example, on [0, 20], P(5 ≤ X ≤ 10) = 5 ÷ 20 = 0.25.

What is the variance of a uniform distribution?

For a continuous uniform on [a, b], variance is (b − a)² ÷ 12. For a discrete uniform on the integers 1 to n, variance is (n² − 1) ÷ 12. Take the square root for standard deviation.

Do I need a calculator for uniform distribution questions?

Usually basic arithmetic is enough. You may need the square root key on the TI BA II Plus or HP 12C for standard deviation, for example 12 then the √x key.