IAI Actuarial Core Principles · Actuarial Statistics

Random Sampling and Sampling Distributions for IAI CS1

Random sampling and sampling distributions describe how a statistic, such as the sample mean, behaves across repeated samples from a population. You solve questions by naming the statistic, finding its distribution (exact, or approximate by the Central Limit Theorem), then computing the probability or quantile from that distribution.

What this chapter covers

This chapter moves you from a single data set to the idea that a statistic is itself a random variable. You start with the population, the sample and what makes a sample random. You then study the sample mean and sample variance, and ask how they vary from one sample to the next.

The core of the chapter is the sampling distribution. For a normal population, the sample mean is exactly normal, and (n − 1)S² ÷ σ² follows a chi-square distribution with n − 1 degrees of freedom. For other populations, the Central Limit Theorem gives an approximate normal result for large n. The chi-square, t and F distributions come from these results, and you use them for inference. The chapter ends with order statistics and sample quantiles, which are the basis of the median, the range and quantile-based summaries.

This chapter feeds directly into the rest of CS1. Estimation, confidence intervals, hypothesis tests, regression and Bayesian statistics all rely on sampling distributions. If this chapter is weak, those later chapters feel like memorised recipes. If it is strong, they become applications of the same few ideas. The same thinking also supports CS2 and the R-based Paper B, where you simulate samples and check distributions.

Sampling distributions sit under statistical inference, which carries a large share of the 2026 CS1 syllabus, and under regression too. Questions here reward clear reasoning: state the distribution, state the assumption, then compute. Many MCQs test one rule, such as the variance of a sample mean or the degrees of freedom of a t statistic, so they are quick marks if you know the conditions. Written questions often ask you to derive or justify a distribution, and you lose marks if you skip the assumption. Time spent here also saves time in every later CS1 chapter, and it helps in Paper B, where you simulate and plot sampling behaviour in R.

Random sampling and sampling distributions: topics in the order to study them

1Population, Sample and Random SamplingYou need the vocabulary and the independence assumption first, because every later result relies on i.i.d. samples.
2Sample Mean and Sample VarianceThese are the main statistics you will study, and you must know their means, variances and the n − 1 divisor before finding their distributions.
3Sampling Distributions and Central Limit TheoremThis gives the exact normal result and the large-sample approximation, which sets up everything that follows.
4Chi-square, t and F DistributionsThese distributions are built from the normal results above, so they come once the sample mean and variance results are clear.
5Order Statistics and Sample QuantilesThis is a separate toolkit that uses the same i.i.d. setup, so it is best learned last when the main ideas are secure.

How to prepare Random sampling and sampling distributions

Aim to understand where each distribution comes from, then practise applying it under exam conditions. Short, repeated sessions work well if you study on a phone alongside work.

Write the i.i.d. setup and the notation (X̄, S², μ, σ²) on one page, and keep it as your reference.
Learn the mean and variance of X̄ and the unbiasedness of S². Check them by deriving them once from the definitions.
Practise Central Limit Theorem questions. Always state n, the mean and the variance of X̄ before you standardise, and say that the result is approximate.
Build the chi-square, t and F results from the normal results. Write each definition as a ratio or sum of squares, with its degrees of freedom and its assumptions.
Practise the order statistics formulas for the maximum and minimum of i.i.d. variables, using the distribution function, then differentiate for the density.
Do past-style MCQs for speed, then written questions for method. In Paper B practice, simulate samples in R and compare them with the theory.

Common mistakes in Random sampling and sampling distributions

Using σ² instead of σ² ÷ n as the variance of the sample mean.
Fix: Write Var(X̄) = σ² ÷ n as your first line, then standardise with the standard deviation σ ÷ √n.
Applying the Central Limit Theorem as if it were exact, or to a small sample from a skewed population.
Fix: Say the result is approximate and for large n. If the population is normal, use the exact result instead.
Using the wrong degrees of freedom, such as n instead of n − 1 for the t or chi-square statistic.
Fix: Whenever X̄ replaces μ in the sum of squares, use n − 1. Write the degrees of freedom next to the distribution name.
Using a z value when σ is unknown and the sample is small.
Fix: If you use S in place of σ with a normal population, use the t distribution with n − 1 degrees of freedom.
Forgetting the independence assumption between X̄ and S², or between the two chi-square variables in an F ratio.
Fix: State the assumptions in one line: normal population, independent samples. Examiners award marks for this.
Getting the order statistics wrong, for example using F(x) instead of [F(x)]ⁿ for the maximum.
Fix: Start from P(max ≤ x) = P(all Xᵢ ≤ x) = [F(x)]ⁿ, then differentiate if you need the density.

Last-day revision: Random sampling and sampling distributions

A statistic is a function of the sample, so it is a random variable with its own distribution.
For an i.i.d. sample with mean μ and variance σ²: E(X̄) = μ and Var(X̄) = σ² ÷ n.
The sample variance S² = Σ(Xᵢ − X̄)² ÷ (n − 1) is an unbiased estimator of σ².
For a normal population, X̄ ~ N(μ, σ² ÷ n) exactly, for any n.
For a normal population, (n − 1)S² ÷ σ² ~ χ² with n − 1 degrees of freedom, and X̄ and S² are independent.
The Central Limit Theorem: for i.i.d. variables with finite variance, (X̄ − μ) ÷ (σ ÷ √n) is approximately N(0, 1) for large n.
If Z ~ N(0, 1) and V ~ χ²(k) are independent, then Z ÷ √(V ÷ k) ~ t with k degrees of freedom.
For a normal sample with unknown σ, (X̄ − μ) ÷ (S ÷ √n) ~ t with n − 1 degrees of freedom.
If U ~ χ²(m) and V ~ χ²(k) are independent, then (U ÷ m) ÷ (V ÷ k) ~ F with (m, k) degrees of freedom.
For i.i.d. X with distribution function F: P(max ≤ x) = [F(x)]ⁿ and P(min > x) = [1 − F(x)]ⁿ.
Sample quantiles come from the ordered sample. Check the interpolation rule your question or R uses.

Random sampling and sampling distributions practice questions

Random sampling and sampling distributions in other exams

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

Random sampling and sampling distributions: frequently asked questions

How is this chapter linked to the rest of CS1?

It supplies the distributions used in estimation, confidence intervals, hypothesis tests and regression. Standard errors, t tests and F tests all rely on the results here. It is worth learning well before you move on.

Do I need to prove the sampling distribution results?

Know the derivations for the mean and variance of X̄ and the unbiasedness of S², as these can be asked in written questions. For the chi-square, t and F results, know the definitions and the conditions well enough to justify a distribution in your own words.

When do I use the t distribution instead of the normal?

Use t when the population is normal, σ is unknown and you replace it with S. The statistic (X̄ − μ) ÷ (S ÷ √n) then has a t distribution with n − 1 degrees of freedom. If σ is known, use the normal.

How does this chapter appear in the computer-based paper?

Paper B is computer-based and uses R. Expect to simulate repeated samples, compute X̄ or S² for each, and compare the results with the theory. Practise this alongside the theory so that the two reinforce each other.