IFoA CS1 common mistakes
IFoA CS1 · Understand the traps and how to avoid them.
Writing the likelihood as a probability density value without the product over the sample, then differentiating a single observation's density to find the MLE.
Candidates learn the word 'likelihood' alongside the density f(x;θ) and conflate the two, especially when the sample is written as a single generic x rather than x1,...,xn. Under exam pressure the product sign is dropped.
What to do instead
Always define L(θ) = ∏_{i=1}^{n} f(x_i;θ) explicitly on the page before differentiating, and if it is simpler, work with l(θ)=log L(θ). Check the result is a function of the data summary (e.g. x̄ or ∑x_i), not of a single x; if a lone x survives, the product was lost.
Taking second derivatives of the log-likelihood and declaring a maximum whenever the second derivative is negative at the stationary point, without checking the parameter space boundary or verifying the stationary point lies inside it.
Standard calculus reflex is d²l/dθ² < 0 ⇒ maximum, but for many CS1 models (e.g. uniform, or constrained parameters) the MLE sits on the boundary where the derivative never vanishes, and for multi-parameter models the Hessian must be negative definite, not just diagonal entries negative.
What to do instead
After solving dl/dθ=0, state the parameter space, confirm the solution is interior, and if not, analyse the likelihood on the boundary directly. For two parameters, evaluate the full Hessian determinant and the leading principal minor, or argue monotonicity of l(θ) instead.
Computing a confidence interval or test statistic with the wrong standard error because the variance of the estimator was taken as Var(X)/n when the parameterisation gives something else (e.g. using σ²/n for a Poisson mean, or confusing the variance of the sample mean with the variance of a single observation).
Candidates memorise 'divide by n' as a universal rule and do not re-derive Var(θ̂) from the model. In Poisson, Var(X)=λ so Var(X̄)=λ/n; in exponential with mean θ, Var(X̄)=θ²/n; mixing these up is common.
What to do instead
Derive Var(θ̂) from first principles each time: write θ̂ as a function of the X_i, use independence, and substitute the model's own variance. Then plug the estimate in only at the final step, keeping the algebraic form visible for method marks.
Using the wrong degrees of freedom or the wrong tail in chi-square goodness-of-fit and contingency-table tests, e.g. using k−1 instead of k−1−p for estimated parameters, or (r−1)(c−1) when parameters were estimated from the table.
The rule 'subtract one for the mean' is over-generalised, and candidates forget that each parameter estimated from the data costs an extra degree of freedom. In contingency tables, estimating expected cell probabilities from margins also changes df.
What to do instead
Before looking up the critical value, write down: number of cells k, number of parameters estimated from the data p, df = k−1−p. For contingency tables, state whether margins are fixed (df=(r−1)(c−1)) or parameters estimated, and adjust accordingly. Show the df calculation as a line of working.
Reporting a p-value or conclusion from a hypothesis test while ignoring the direction of the alternative, e.g. doubling or not doubling the tail incorrectly for a one-sided versus two-sided test.
Candidates compute the test statistic, look up a two-tailed critical value out of habit, and then interpret it against a one-sided alternative (or vice versa). The asymmetry between H1: θ>θ0 and H1: θ≠θ0 is easily lost under time pressure.
What to do instead
Write H0 and H1 explicitly at the top, mark the rejection region on a sketch of the null distribution, and only then compute. For a one-sided test the p-value is P(T≥t) or P(T≤t) as appropriate; for two-sided it is 2×min of those. State which you used in the conclusion.
Confusing the probability generating function, moment generating function and characteristic function, and using the wrong one to find the distribution of a sum or to extract moments.
All three are 'transforms' with similar notation G(t), M(t), φ(t), and candidates mix up which exists for which distribution (e.g. PGF only for non-negative integer-valued variables) and which operation corresponds to summing independent variables.
What to do instead
Keep a one-line table: PGF G(z)=E[z^X] for counts, MGF M(t)=E[e^{tX}] generally, CF φ(t)=E[e^{itX}] always. For sums of independent variables multiply the relevant transform; differentiate at the right point (G'(1) for mean of counts, M'(0) for mean generally). Use the transform valid for the support.
Applying a normal approximation to a sum or average without a continuity correction when the underlying variable is discrete, or applying it when n is too small for the approximation to be credible.
The central limit theorem is drilled as 'use normal for large n', and candidates drop the continuity correction because it feels like a technicality; they also fail to check that np and n(1−p) are adequate.
What to do instead
For a discrete sum S, approximate P(S≤s) by P(S≤s+0.5) and P(S≥s) by P(S≥s−0.5) using the normal with the correct mean and variance. State the validity check (e.g. np>5 and n(1−p)>5, or n≥30 for means) before using the approximation.
Setting up a Markov chain transition matrix with rows that do not sum to one, or ordering states inconsistently between the matrix and the initial distribution vector.
Candidates write transition probabilities from a word problem in the order the states are mentioned rather than a declared order, and rounding in multi-step calculations leaves rows summing to 0.99 or 1.01 without being noticed.
What to do instead
Declare the state ordering explicitly (e.g. states 1,2,3) and label rows and columns before filling entries. After writing P, verify every row sums to 1. When computing P^n or πP, keep the same ordering and check the resulting vector sums to 1.
Computing a compound or aggregate claim distribution by adding means and variances of the wrong quantities, e.g. treating the sum of a random number of claims as if the number were fixed, or mixing up the conditional variance formula.
The law of total variance Var(S)=E[N]Var(X)+Var(N)(E[X])² is easy to state but easy to misapply; candidates often use Var(S)=N Var(X) with N fixed, or forget the second term entirely.
What to do instead
Write S=∑_{i=1}^{N} X_i with N random and independent of the X_i, then apply E[S]=E[N]E[X] and Var(S)=E[N]Var(X)+Var(N)(E[X])². Substitute numbers only after the symbolic formula is written, and sanity-check that Var(S) exceeds E[N]Var(X).
Losing marks on the written interpretation: giving a numerical answer with no statement in context, or claiming a confidence interval is 'the probability the parameter lies in the interval'.
Candidates focus on the algebra and treat the final sentence as optional, and the frequentist interpretation of confidence is genuinely counter-intuitive, so the incorrect phrasing is repeated from casual usage.
What to do instead
End every estimation or test question with one sentence in context, e.g. 'We are 95% confident that the true mean claim size lies between a and b' or 'There is insufficient evidence at the 5% level to reject H0'. Never say the parameter has a probability of lying in a fixed interval; describe the procedure's long-run coverage instead.