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

Foundation Mathematics, Statistics and Data Science (CS1, CS2, CM1 foundations) syllabus: all 4 chapters

Expand any chapter to see its topics and sub-topics.

1. 1

   Probability and Mathematical Statistics

   4 topics

   • Random variables and distributions
     • Discrete distributions: binomial, Poisson, geometric, negative binomial, hypergeometric
     • Continuous distributions: exponential, gamma, beta, normal, lognormal, chi-square, t, F
     • Moments, moment generating functions and cumulant generating functions
     • Conditional and joint distributions, covariance and correlation
   • Sampling distributions and estimation
     • Central Limit Theorem and sampling distribution of the mean
     • Method of moments and maximum likelihood estimation
     • Unbiasedness, efficiency, consistency and Cramer-Rao lower bound
     • Confidence intervals for means, proportions and variances
   • Hypothesis testing
     • Type I and Type II errors, power and p-values
     • Tests for means and variances (z, t, chi-square, F tests)
     • Chi-square goodness-of-fit and contingency tables
   • Bayesian statistics
     • Prior, likelihood and posterior distributions
     • Conjugate priors and credibility theory link
     • Loss functions and Bayesian estimators
2. 2

   Regression and Statistical Modelling

   3 topics

   • Linear regression
     • Simple and multiple linear regression assumptions
     • Least squares estimation and inference on coefficients
     • Residual analysis, leverage and influence diagnostics
   • Generalised linear models (GLMs)
     • Exponential family and link functions
     • Poisson, binomial and gamma GLMs
     • Deviance, scaled deviance and model comparison
   • Analysis of variance
3. 3

   Stochastic Processes and Time Series

   4 topics

   • Markov chains
     • Transition matrices, Chapman-Kolmogorov equations
     • Stationary distributions and classification of states
   • Markov jump processes
     • Poisson process and inter-event times
     • Kolmogorov forward and backward equations
     • Time-homogeneous and time-inhomogeneous models
   • Time series analysis
     • Stationarity, autocovariance and autocorrelation functions
     • ARIMA model identification and estimation
     • Forecasting and the Box-Jenkins methodology
   • Brownian motion and diffusion processes
4. 4

   Survival Models and Machine Learning

   3 topics

   • Survival analysis
     • Lifetime distributions, hazard and survival functions
     • Estimation of lifetime distributions and the Cox proportional hazards model
     • Kaplan-Meier and Nelson-Aalen estimators with censoring
   • Mortality estimation
     • Central and initial exposed to risk
     • Graduation methods and statistical tests of fit
   • Introduction to machine learning
     • Supervised vs unsupervised learning
     • Cross-validation, bias-variance trade-off and regularisation
     • Decision trees, random forests and neural network basics

Institute and Faculty of Actuaries (IFoA) Exams Foundation Mathematics, Statistics and Data Science (CS1, CS2, CM1 foundations) flashcards — 50 questions with answers

1. State the relationship between the probability density function (pdf) and the cumulative distribution function (cdf) of a continuous random variable $X$.

   The cdf is $F(x)=\int_{-\infty}^{x} f(t)\,dt$, and conversely the pdf is the derivative $f(x)=\frac{d}{dx}F(x)$ wherever $F$ is differentiable.

2. Define the moment generating function (MGF) of a random variable $X$ and state how it generates the $n$th moment.

   $M_X(t)=E[e^{tX}]$. The $n$th raw moment is obtained by differentiating and setting $t=0$: $E[X^{n}]=M_X^{(n)}(0)$.

3. For a Poisson random variable with parameter $\lambda$, state its probability function, mean and variance.

   $P(X=x)=\frac{e^{-\lambda}\lambda^{x}}{x!}$ for $x=0,1,2,\dots$, with $E[X]=\lambda$ and $\operatorname{Var}(X)=\lambda$.

4. State the mean and variance of a Gamma distribution with shape $\alpha$ and rate $\lambda$.

   $E[X]=\frac{\alpha}{\lambda}$ and $\operatorname{Var}(X)=\frac{\alpha}{\lambda^{2}}$. Its pdf is $f(x)=\frac{\lambda^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x}$, $x>0$.

5. State the Central Limit Theorem for the sample mean $\bar{X}$ of $n$ iid variables with mean $\mu$ and variance $\sigma^{2}$.

   As $n\to\infty$, $\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\xrightarrow{d}N(0,1)$; equivalently $\bar{X}\approx N\!\left(\mu,\frac{\sigma^{2}}{n}\right)$ for large $n$ regardless of the parent distribution.

6. What distribution does $\frac{(n-1)S^{2}}{\sigma^{2}}$ follow when sampling from a normal population, and why is it useful?

   It follows a chi-square distribution with $n-1$ degrees of freedom, $\chi^{2}_{n-1}$. This underpins confidence intervals and tests for the population variance $\sigma^{2}$.

7. Define an unbiased estimator and state the property of efficiency for two unbiased estimators.

   An estimator $\hat{\theta}$ is unbiased if $E[\hat{\theta}]=\theta$. Among unbiased estimators, the more efficient one has the smaller variance; the minimum is bounded below by the Cramer-Rao lower bound.

8. Describe the method of maximum likelihood estimation (MLE).

   Form the likelihood $L(\theta)=\prod_i f(x_i;\theta)$, take the log-likelihood $\ell(\theta)=\ln L(\theta)$, then solve $\frac{d\ell}{d\theta}=0$ (checking it is a maximum) to obtain $\hat{\theta}$.

9. Give the general form of a two-sided $100(1-\alpha)\%$ confidence interval for a normal mean $\mu$ when $\sigma$ is unknown.

   $\bar{x}\pm t_{n-1,\,1-\alpha/2}\,\frac{s}{\sqrt{n}}$, where $s$ is the sample standard deviation and $t_{n-1,\,1-\alpha/2}$ is the relevant $t$-quantile.

10. Define the significance level and the power of a hypothesis test, and relate power to the Type II error.

    The significance level $\alpha=P(\text{reject }H_0\mid H_0\text{ true})$ (Type I error). Power $=1-\beta$, where $\beta=P(\text{accept }H_0\mid H_0\text{ false})$ is the Type II error probability.

11. What is a $p$-value, and what is the decision rule using it?

    The $p$-value is the probability, assuming $H_0$ is true, of obtaining a test statistic at least as extreme as that observed. Reject $H_0$ if $p\text{-value}<\alpha$.

12. State the Neyman-Pearson lemma in words.

    For testing a simple null against a simple alternative, the likelihood ratio test that rejects $H_0$ when $\frac{L(\theta_0)}{L(\theta_1)}\leq k$ is the most powerful test of its size $\alpha$.

13. What test statistic is used for a chi-square goodness-of-fit test, and its approximate distribution?

    $\chi^{2}=\sum_i \frac{(O_i-E_i)^{2}}{E_i}$, where $O_i$ and $E_i$ are observed and expected frequencies. Under $H_0$ it is approximately $\chi^{2}$ with degrees of freedom = (categories $-1-$ parameters estimated).

14. State Bayes' theorem for a parameter $\theta$ given data $x$.

    $f(\theta\mid x)=\frac{f(x\mid\theta)\,f(\theta)}{\int f(x\mid\theta)\,f(\theta)\,d\theta}\propto f(x\mid\theta)\,f(\theta)$, i.e. posterior $\propto$ likelihood $\times$ prior.

15. Define a conjugate prior and give one example.

    A prior is conjugate if the posterior belongs to the same family. Example: a Beta$(\alpha,\beta)$ prior with binomial data gives a Beta$(\alpha+x,\beta+n-x)$ posterior.

16. Under quadratic (squared-error) loss, what is the Bayes estimator of $\theta$?

    The posterior mean, $E[\theta\mid x]$. Under absolute-error loss it is the posterior median, and under all-or-nothing (0/1) loss it is the posterior mode.

17. Describe how credibility theory's Bayesian estimate is a weighted average.

    The credibility estimate is $Z\bar{X}+(1-Z)\mu$, a weighted average of the observed sample mean $\bar{X}$ and the prior/collective mean $\mu$, where $Z\in[0,1]$ is the credibility factor.

18. Write the simple linear regression model and the least-squares slope estimator.

    Model: $Y_i=\alpha+\beta x_i+\varepsilon_i$ with $\varepsilon_i\sim N(0,\sigma^{2})$ iid. Slope: $\hat{\beta}=\frac{S_{xy}}{S_{xx}}=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^{2}}$, intercept $\hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}$.

19. Define the coefficient of determination $R^{2}$ and its interpretation.

    $R^{2}=\frac{SS_{\text{reg}}}{SS_{\text{tot}}}=1-\frac{SS_{\text{res}}}{SS_{\text{tot}}}$. It is the proportion of total variation in $Y$ explained by the regression, with $0\leq R^{2}\leq 1$.

20. List the four standard assumptions of the linear regression error term.

    Errors $\varepsilon_i$ have (1) zero mean, (2) constant variance $\sigma^{2}$ (homoscedasticity), (3) independence, and (4) normality. Equivalently $\varepsilon_i\sim N(0,\sigma^{2})$ iid.

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