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 — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Foundation Mathematics, Statistics and Data Science (CS1, CS2, CM1 foundations) in Institute and Faculty of Actuaries (IFoA) Exams, not a summary of it.

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

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

21 of 50 cards from the Foundation Mathematics, Statistics and Data Science (CS1, CS2, CM1 foundations) deck — real questions with worked 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.

21. State the three components that define a generalised linear model (GLM).

    (1) A random component: response from the exponential family. (2) A systematic component: linear predictor $\eta=\sum_j \beta_j x_j$. (3) A link function $g$ with $g(\mu)=\eta$, connecting the mean $\mu$ to the linear predictor.

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