Institute and Faculty of Actuaries (IFoA) Exams Foundation Mathematics, Statistics and Data Science (CS1, CS2, CM1 foundations) Flashcards

24 sample cards from the Foundation Mathematics, Statistics and Data Science (CS1, CS2, CM1 foundations) deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Define a Markov chain and the Markov (memoryless) property.

   A stochastic process $\{X_n\}$ on a discrete state space with the Markov property: $P(X_{n+1}=j\mid X_n=i, X_{n-1},\dots)=P(X_{n+1}=j\mid X_n=i)$. The future depends only on the present state, not the past.

2. State the Chapman-Kolmogorov equations for an $n$-step transition matrix.

   $P^{(m+n)}=P^{(m)}P^{(n)}$, i.e. $p_{ij}^{(m+n)}=\sum_k p_{ik}^{(m)}p_{kj}^{(n)}$. In particular the $n$-step matrix is $P^{(n)}=P^{n}$, the $n$th power of the one-step matrix.

3. How is the stationary (equilibrium) distribution $\pi$ of a Markov chain found?

   Solve $\pi P=\pi$ subject to $\sum_i \pi_i=1$, where $P$ is the transition matrix. $\pi$ is the long-run proportion of time spent in each state for an irreducible aperiodic chain.

4. Classify Markov chain states: define recurrent vs transient.

   A state is recurrent if the process returns to it with probability 1 (visited infinitely often); it is transient if there is positive probability of never returning. Recurrent states are positive recurrent if the mean return time is finite, else null recurrent.

5. For a Markov jump process, what distribution governs the holding (waiting) time in a state $i$?

   An exponential distribution with rate equal to the total transition rate out of $i$, namely $\lambda_i=-\sigma_{ii}=\sum_{j\neq i}\sigma_{ij}$, so the mean holding time is $\frac{1}{\lambda_i}$.

6. State the Kolmogorov forward (Fokker-Planck) differential equation for a Markov jump process.

   $\frac{d}{dt}P(t)=P(t)A$, where $P(t)$ is the matrix of transition probabilities and $A$ is the generator (transition-rate) matrix. The backward equation is $\frac{d}{dt}P(t)=A\,P(t)$.

7. What property defines the generator matrix $A$ of a continuous-time Markov chain?

   Off-diagonal entries $\sigma_{ij}\geq 0$ are transition rates $i\to j$; each row sums to zero, so $\sigma_{ii}=-\sum_{j\neq i}\sigma_{ij}\leq 0$.

8. Define a (weakly / covariance) stationary time series.

   A series $\{X_t\}$ is weakly stationary if $E[X_t]=\mu$ is constant, $\operatorname{Var}(X_t)$ is constant, and the autocovariance $\operatorname{Cov}(X_t,X_{t+k})$ depends only on the lag $k$, not on $t$.

9. Write the model equation for an $AR(p)$ and an $MA(q)$ process.

   $AR(p)$: $X_t=\mu+\sum_{i=1}^{p}\alpha_i(X_{t-i}-\mu)+e_t$. $MA(q)$: $X_t=\mu+e_t+\sum_{j=1}^{q}\beta_j e_{t-j}$, where $\{e_t\}$ is white noise.

10. What identifies the order of pure $AR$ and pure $MA$ models from the ACF and PACF?

    An $MA(q)$ has an ACF that cuts off after lag $q$ (PACF tails off). An $AR(p)$ has a PACF that cuts off after lag $p$ (ACF tails off / decays exponentially).

11. State the stationarity condition for an $AR(1)$ process $X_t=\alpha X_{t-1}+e_t$.

    The process is stationary if and only if $|\alpha|<1$. Then $E[X_t]=0$, $\operatorname{Var}(X_t)=\frac{\sigma^{2}}{1-\alpha^{2}}$, and the ACF is $\rho_k=\alpha^{|k|}$.

12. List the four defining properties of standard Brownian motion $\{B_t\}$.

    (1) $B_0=0$; (2) independent increments; (3) stationary increments with $B_t-B_s\sim N(0,\,t-s)$ for $s<t$; (4) continuous sample paths. It has mean $0$ and variance $t$.

13. Write the stochastic differential equation (SDE) for a general diffusion / Ito process.

    $dX_t=\mu(X_t,t)\,dt+\sigma(X_t,t)\,dB_t$, where $\mu$ is the drift coefficient and $\sigma$ the diffusion (volatility) coefficient, and $B_t$ is standard Brownian motion.

14. State Ito's lemma for $f(X_t)$ where $dX_t=\mu\,dt+\sigma\,dB_t$.

    $df=\left(\mu\,\frac{\partial f}{\partial x}+\frac{1}{2}\sigma^{2}\,\frac{\partial^{2} f}{\partial x^{2}}\right)dt+\sigma\,\frac{\partial f}{\partial x}\,dB_t$. The extra second-derivative term arises from $(dB_t)^{2}=dt$.

15. Define geometric Brownian motion and give the distribution of $X_t$.

    $dX_t=\mu X_t\,dt+\sigma X_t\,dB_t$, with solution $X_t=X_0\exp\!\big((\mu-\tfrac{1}{2}\sigma^{2})t+\sigma B_t\big)$. Thus $\ln X_t$ is normal, so $X_t$ is lognormally distributed.

16. Define the survival function $S(t)$ and the hazard rate $\mu(t)$ in survival analysis.

    $S(t)=P(T>t)=1-F(t)$, the probability of surviving beyond $t$. The hazard rate is $\mu(t)=\frac{f(t)}{S(t)}=-\frac{d}{dt}\ln S(t)$, the instantaneous failure rate given survival to $t$.

17. State the relationship between the survival function and the integrated hazard.

    $S(t)=\exp\!\left(-\int_0^{t}\mu(s)\,ds\right)=e^{-\Lambda(t)}$, where $\Lambda(t)=\int_0^t\mu(s)\,ds$ is the cumulative (integrated) hazard.

18. What is censoring in survival data, and distinguish right-censoring from left-censoring.

    Censoring occurs when the exact event time is unknown, only bounded. Right-censoring: the event has not occurred by the end of observation ($T>c$). Left-censoring: the event occurred before observation began ($T<c$).

19. Write the Cox proportional hazards model and state its key assumption.

    $\mu(t;x)=\mu_0(t)\exp(\beta^{\top}x)=\mu_0(t)\exp(\beta_1 x_1+\cdots+\beta_p x_p)$. Key assumption: hazards for different covariate values are proportional, i.e. their ratio is constant over time, and $\mu_0(t)$ is an unspecified baseline hazard.

20. Define the central rate of mortality $m_x$ and the initial rate $q_x$.

    $q_x=P(\text{life aged }x\text{ dies within a year})$, with central exposure denominator. The central rate is $m_x=\frac{\text{deaths aged }x}{\text{central exposed to risk}}$; approximately $m_x\approx\frac{q_x}{1-\frac{1}{2}q_x}$ under uniform deaths.

21. What is the maximum likelihood estimator of the constant force of mortality $\mu$ in the two-state model?

    $\hat{\mu}=\frac{D}{E_c^{x}}$, the number of observed deaths $D$ divided by the central exposed to risk (total waiting time) $E_c^{x}$. Its variance is estimated by $\frac{D}{(E_c^{x})^{2}}$.

22. Distinguish supervised from unsupervised learning in machine learning.

    Supervised learning uses labelled data to learn a mapping from inputs to a known output (e.g. regression, classification). Unsupervised learning finds structure in unlabelled data (e.g. clustering, dimension reduction).

23. Explain the bias-variance trade-off in predictive modelling.

    Expected test error decomposes as $\text{bias}^{2}+\text{variance}+\text{irreducible error}$. Simple models have high bias/low variance (underfit); complex models have low bias/high variance (overfit). The goal is to minimise total error.

24. What is $k$-fold cross-validation and why is it used?

    The data are split into $k$ folds; the model is trained on $k-1$ folds and validated on the remaining fold, repeated $k$ times and averaged. It gives a more reliable estimate of out-of-sample (generalisation) error and helps tune model complexity.

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