IAI Actuarial Core Principles · Actuarial Statistics

Expectations and Conditional Expectations for IAI Actuarial Statistics

Expectation is the probability-weighted average of a random variable: E[X] = Σ x P(X = x) for discrete, ∫ x f(x) dx for continuous. Conditional expectation E[X | Y] averages X given Y. To solve questions, find the conditional distribution, then use E[X] = E[E[X | Y]] and Var(X) = E[Var(X | Y)] + Var(E[X | Y]).

What this chapter covers

This chapter covers the average value of a random variable and how that average changes when you know something else. You start with the definition of expectation for discrete and continuous variables, including E[g(X)]. You then build the toolkit: linearity, variance, covariance and moments.

The second half deals with conditional distributions and conditional expectation. Here E[X | Y] is itself a random variable, a function of Y. Two results follow from this idea: the law of total expectation, E[X] = E[E[X | Y]], and the law of total variance, Var(X) = E[Var(X | Y)] + Var(E[X | Y]).

This chapter supports much of the paper. Compound distributions in CS2 risk modelling, mixture models, credibility and Bayesian statistics in CS1 all use conditioning. Regression is also conditional expectation: the model for E[Y | X]. If you are weak here, those later chapters will feel harder than they should.

Questions from this chapter appear both as short multiple-choice items and as multi-step written questions, and the same tools sit inside questions from other chapters, such as Bayesian statistics, regression and compound distributions. The calculations are short once you know the pattern, so they are good marks for the effort. They also reward clean setup, which written answers need: define the variables, state the rule, then compute. Master the two total laws and you can often solve a mixture or compound problem without finding the full distribution.

Expectations and conditional expectations: topics in the order to study them

1Expectation of a Random VariableEverything else rests on the definition of E[X] and E[g(X)], so you must be fluent in the sums and integrals first.
2Properties of Expectation, Variance and MomentsLinearity, variance rules and covariance let you shortcut calculations, and you need them for conditional work.
3Conditional Distributions and Conditional ExpectationYou need conditional pmfs and pdfs and the idea that E[X | Y] is a random variable before you can use the total laws.
4Law of Total Expectation and Total VarianceThis is the payoff topic. It combines all earlier ideas and is the form most exam questions take.

How to prepare Expectations and conditional expectations

Build the chapter in layers. Each layer should be automatic before you move to the next, and you should write out working every time.

Write down the definitions of E[X], E[g(X)], Var(X) = E[X²] − (E[X])² and Cov(X, Y) = E[XY] − E[X]E[Y]. Practise computing them from a small pmf and from a simple pdf.
Learn the rules with their conditions: E[aX + b] = aE[X] + b always holds; Var(aX + b) = a²Var(X); Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y), and the covariance term vanishes if X and Y are independent.
Practise getting conditional distributions from a joint pmf or pdf: f(x | y) = f(x, y) ÷ f_Y(y), for y where f_Y(y) > 0. Then compute E[X | Y = y] directly.
Treat E[X | Y] as a function of Y. Write it as g(Y), then take its expectation and variance. Do this until it feels natural.
Apply the two total laws to mixtures and compound models, for example a Poisson count whose mean depends on another variable. Write out each of the two variance terms separately.
Do past-paper questions under time. Show the setup, formula, working and result. Check the answer with a quick sanity test, such as whether the variance is non-negative.
Revisit the chapter after a few days and redo one question from each topic without notes.

Common mistakes in Expectations and conditional expectations

Writing Var(X) = E[E[Var(X | Y)]] only, leaving out the second term of the total variance.
Fix: Always write both parts: E[Var(X | Y)] + Var(E[X | Y]). Compute each separately and add.
Using E[g(X)] = g(E[X]) for non-linear g, such as E[X²] = (E[X])².
Fix: Only linear functions pass through the expectation. For X², use E[X²] = Var(X) + (E[X])².
Treating zero covariance as proof of independence, or adding variances when variables are dependent.
Fix: State whether independence is given. If not, include the 2Cov(X, Y) term.
Dividing by the wrong marginal when finding a conditional density, or forgetting to integrate over the correct range.
Fix: Write f(x | y) = f(x, y) ÷ f_Y(y) explicitly. Find the marginal first, and check that the conditional density integrates to 1.
Treating E[X | Y] as a fixed number when asked for its variance or expectation.
Fix: Write E[X | Y] = g(Y) as a function of Y, then apply expectation or variance to g(Y).
Leaving out the setup in written answers and giving only a final number.
Fix: Define the variables, state the rule you use, show the working and give the result. Method marks depend on it.

Last-day revision: Expectations and conditional expectations

E[X] = Σ x P(X = x) for discrete; E[X] = ∫ x f(x) dx for continuous.
E[g(X)] = Σ g(x) P(X = x) or ∫ g(x) f(x) dx. In general, E[g(X)] ≠ g(E[X]).
Linearity: E[aX + bY] = aE[X] + bE[Y], with no independence needed.
Var(X) = E[X²] − (E[X])²; Var(aX + b) = a²Var(X).
Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y); if independent, Cov = 0.
Zero covariance does not imply independence.
If X and Y are independent, E[XY] = E[X]E[Y].
Conditional pdf: f(x | y) = f(x, y) ÷ f_Y(y), where f_Y(y) > 0.
E[X | Y] is a random variable, a function of Y.
Law of total expectation: E[X] = E[E[X | Y]].
Law of total variance: Var(X) = E[Var(X | Y)] + Var(E[X | Y]).
For compound sums S = X₁ + … + X_N with i.i.d. claims independent of N: E[S] = E[N]E[X] and Var(S) = E[N]Var(X) + Var(N)(E[X])².

Expectations and conditional expectations practice questions

Expectations and conditional expectations in other exams

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

FRM Part IFundamentals of Probability

Expectations and conditional expectations: frequently asked questions

What is the difference between E[X | Y = y] and E[X | Y]?

E[X | Y = y] is a number: the mean of X when Y takes the value y. E[X | Y] is a random variable, because it is that mean viewed as a function of Y. You take expectations and variances of E[X | Y] in the total laws.

When do I use the law of total variance?

Use it when X depends on another variable Y and you know the conditional mean and variance of X given Y. It splits the variance into the average within-group variance and the variance of the group means. It is common for mixtures and compound distributions.

Is E[XY] = E[X]E[Y] always true?

No. It holds if X and Y are independent, and more generally if they are uncorrelated. Otherwise E[XY] = E[X]E[Y] + Cov(X, Y).

How does this chapter link to regression and Bayesian statistics?

A regression model describes E[Y | X] as a function of the explanatory variables. In Bayesian work you condition on data to get a posterior, and its mean is a conditional expectation. The same rules apply in both.