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Actuarial Statistics · Basic univariate distributions and generating samples

Uniform Distribution and Distribution Properties for IAI Actuarial Statistics

Updated 11 October 2026 · Fact-checked

The uniform distribution gives equal probability to every value in a range. For continuous U(a, b), the mean is (a + b) ÷ 2 and the variance is (b − a)² ÷ 12. Distribution properties link families: exponential is gamma with α = 1, and chi-square with ν degrees of freedom is gamma(ν/2, 1/2). Look up the rest in the Tables.

Understand Uniform Distribution and Distribution Properties

A uniform distribution spreads probability evenly. Nothing in the range is more likely than anything else. That makes it the simplest model and the base for simulation, because computers first generate U(0, 1) numbers and then convert them to other distributions.

The continuous uniform on (a, b) has a flat density f(x) = 1 ÷ (b − a) for a < x < b. The area under the flat line must equal 1, so the height is 1 ÷ (b − a). Probabilities are just lengths: P(c < X < d) = (d − c) ÷ (b − a), for a ≤ c ≤ d ≤ b. The discrete uniform on the integers 1, 2, ..., n gives each value probability 1/n. Think of a fair die with n = 6.

Many distributions are related. The exponential with rate λ is a special case of the gamma: gamma(1, λ). A sum of n independent exponential(λ) variables is gamma(n, λ). A chi-square with ν degrees of freedom is gamma(ν/2, 1/2). The square of a standard normal is chi-square with 1 degree of freedom. If you know one distribution, you can often get another without new work.

The normal approximation to the binomial works when n is large and p is not too close to 0 or 1. A common rule of thumb is that np and n(1 − p) should both be reasonably large (often at least 5 or 10). A binomial is discrete and the normal is continuous, so you apply a continuity correction: treat the integer k as the interval from k − 0.5 to k + 0.5.

The Formulae and Tables for the Actuarial Examinations list densities, means, variances and moment generating functions of standard distributions. You do not need to memorise most of them. You do need to know how to read them, notice the parameterisation used (for example gamma with α and λ), and apply them correctly. The Tables also contain the standard normal and chi-square percentage points.

Key rules to remember

Continuous uniform U(a, b) density and CDF
f(x) = 1 ÷ (b − a) for a < x < b; F(x) = (x − a) ÷ (b − a) for a ≤ x ≤ b
Outside (a, b) the density is 0. Probabilities are lengths divided by (b − a).
Continuous uniform mean and variance
E[X] = (a + b) ÷ 2; Var(X) = (b − a)² ÷ 12
The mean is the midpoint. The variance depends only on the width.
Discrete uniform on 1, 2, ..., n
P(X = k) = 1 ÷ n; E[X] = (n + 1) ÷ 2; Var(X) = (n² − 1) ÷ 12
If the values run a, a+1, ..., b, use n = b − a + 1 and shift the mean by a − 1. The variance is unchanged by shifting.
Uniform MGF
M(t) = (e^(bt) − e^(at)) ÷ ((b − a)t), t ≠ 0
M(0) = 1.
Exponential as gamma
Exponential(λ) = Gamma(1, λ)
Gamma(α, λ) has mean α ÷ λ and variance α ÷ λ². Check the Tables for the parameterisation.
Chi-square as gamma
χ²(ν) = Gamma(ν ÷ 2, 1 ÷ 2); mean ν, variance 2ν
Sum of independent χ² variables is χ² with the degrees of freedom added.
Sum of independent exponentials
X₁ + ... + Xₙ ~ Gamma(n, λ) for iid Exponential(λ)
Gives the waiting time to the nth event in a Poisson process.
Normal approximation to binomial
X ~ Bin(n, p) ≈ N(np, np(1 − p))
Use when n is large and p is not near 0 or 1.
Continuity correction
P(X ≤ k) ≈ P(Y < k + 0.5); P(X ≥ k) ≈ P(Y > k − 0.5); P(X = k) ≈ P(k − 0.5 < Y < k + 0.5)
Y is the normal approximation. Apply it for integer-valued X.
Standard normal square
If Z ~ N(0, 1), then Z² ~ χ²(1)
Sum of squares of n independent standard normals is χ²(n).

How to solve Uniform Distribution and Distribution Properties questions

Use this method for any question on uniform distributions, relationships between distributions or approximations.

  1. 1Identify the distribution and write its parameters. State whether it is discrete or continuous.
  2. 2Check the parameterisation in the Formulae and Tables, especially for gamma and exponential (rate versus mean).
  3. 3Write the probability as a length (uniform) or use the CDF or Tables. For a relationship question, rewrite one distribution as the other, for example χ²(ν) as Gamma(ν/2, 1/2).
  4. 4If approximating a discrete variable by a normal, calculate the mean and variance, then apply the continuity correction before standardising.
  5. 5Standardise with z = (x − μ) ÷ σ and read the normal table. Keep at least four decimal places until the end.
  6. 6Check that the answer lies between 0 and 1 and is sensible. State any assumption, such as independence, in words.

Quickest way: Shortcut for uniform and approximation questions

When to use it: Use for multiple-choice questions where you need a fast and reliable answer.

  1. For U(a, b), get the mean as the midpoint and the variance as width² ÷ 12. Do not integrate.
  2. For a uniform probability, draw the interval and divide the length of the favourable part by the total width.
  3. For distribution links, compare the mean and variance with the Tables: gamma(α, λ) has mean α ÷ λ, so exponential has α = 1.
  4. For binomial approximations, write the continuity correction first, as a half-unit shift outward to include the integer.
  5. Check your answer against the extremes: probabilities for a single value of a continuous uniform are 0.

Common mistakes in Uniform Distribution and Distribution Properties

  • Using (b − a)² ÷ 12 as the standard deviation.

    Students remember the formula and forget it is a variance.

    Fix: Take the square root for the standard deviation: (b − a) ÷ √12.

  • Using n = b − a for a discrete uniform on a, ..., b.

    The count of integers is one more than the difference.

    Fix: Use n = b − a + 1 for the number of values.

  • Forgetting the continuity correction, or applying it in the wrong direction.

    Students memorise the rule without thinking of the intervals.

    Fix: Treat each integer k as (k − 0.5, k + 0.5). For P(X ≤ k) go up to k + 0.5. For P(X < k) go to k − 0.5.

  • Mixing up rate and mean for exponential and gamma.

    Some texts use mean θ and others use rate λ.

    Fix: Check the parameterisation in the Tables and confirm by computing the mean.

  • Treating χ²(ν) as Gamma(ν, 1/2) or Gamma(ν/2, 2).

    The halves are easy to misplace.

    Fix: Check: χ²(ν) has mean ν. Gamma(ν/2, 1/2) gives (ν/2) ÷ (1/2) = ν.

  • Using the normal approximation when np is very small.

    Students apply it automatically to any binomial.

    Fix: Check that np and n(1 − p) are both reasonably large. Otherwise compute the binomial exactly or use another approximation.

Worked examples

Example 1

X is uniformly distributed on (10, 40). Find E[X], Var(X) and P(X > 25 | X > 15).

Show the solution
  1. Mean = (10 + 40) ÷ 2 = 25.
  2. Variance = (40 − 10)² ÷ 12 = 900 ÷ 12 = 75.
  3. P(X > 25) = (40 − 25) ÷ 30 = 15 ÷ 30 = 0.5.
  4. P(X > 15) = (40 − 15) ÷ 30 = 25 ÷ 30.
  5. Conditional probability = P(X > 25) ÷ P(X > 15) = 15 ÷ 25 = 0.6.

Answer: E[X] = 25, Var(X) = 75, P(X > 25 | X > 15) = 0.6.

Example 2

X ~ Binomial(100, 0.4). Use a normal approximation with continuity correction to estimate P(X ≤ 45).

Show the solution
  1. Mean = np = 100 × 0.4 = 40.
  2. Variance = np(1 − p) = 100 × 0.4 × 0.6 = 24. Standard deviation = √24 = 4.8990.
  3. With the continuity correction, P(X ≤ 45) ≈ P(Y < 45.5), where Y ~ N(40, 24).
  4. z = (45.5 − 40) ÷ 4.8990 = 1.1227.
  5. Φ(1.12) = 0.8686 and Φ(1.13) = 0.8708, so Φ(1.1227) ≈ 0.8686 + 0.27 × 0.0022 ≈ 0.869.

Answer: P(X ≤ 45) ≈ 0.869.

Exam tips

  • Write the distribution and its parameters at the start. Markers give credit for correct setup even if arithmetic slips.
  • Show the continuity correction line explicitly. Missing it is a frequent source of lost marks.
  • In the Tables, confirm the parameterisation by checking the mean before using any gamma or exponential result.
  • For relationship questions, justify with the density or MGF. For example, matching MGFs proves two distributions are the same.
  • In computer-based Paper B, use R functions such as punif, dunif and runif, and state the parameters clearly in your code comments.

Practice questions from Basic univariate distributions and generating samples

Uniform Distribution and Distribution Properties in other exams

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

Uniform Distribution and Distribution Properties: frequently asked questions

What are the mean and variance of the uniform distribution?

For continuous U(a, b), the mean is (a + b) ÷ 2 and the variance is (b − a)² ÷ 12. For a discrete uniform on 1, ..., n, the mean is (n + 1) ÷ 2 and the variance is (n² − 1) ÷ 12.

How are the gamma, exponential and chi-square distributions related?

The exponential with rate λ is Gamma(1, λ). The chi-square with ν degrees of freedom is Gamma(ν/2, 1/2). A sum of n independent exponential(λ) variables is Gamma(n, λ).

How do I use the Formulae and Tables for distributions?

Find the distribution, check how the parameters are defined, and read off the density, mean, variance and MGF. Use the tables for normal and chi-square percentage points. Practise with them so you do not waste time in the exam.

When do I use the continuity correction?

Use it when approximating a discrete integer-valued variable, such as a binomial or Poisson, by a continuous normal. Replace each integer k by the interval from k − 0.5 to k + 0.5.