Appendix B — Mathematical Notation Reference

Mathematical Notation Reference

This appendix provides a quick reference for mathematical notation used throughout the course. You do not need to memorise these — use this page as a lookup when you encounter unfamiliar symbols.

B.1 Basic notation

Symbol	Meaning	Example
$x$	A variable (lowercase)	Blood pressure measurement
$X$	A random variable (uppercase)	Blood pressure in the population
$\bar{x}$	Sample mean	Mean blood pressure in your study
$\mu$	Population mean	True mean blood pressure
$s$ or $\hat{\sigma}$	Sample standard deviation	SD of blood pressures in your data
$\sigma$	Population standard deviation	True SD of blood pressures
$n$	Sample size	Number of patients
$p$	Probability or proportion	Prevalence of a disease
$\hat{p}$	Estimated proportion	Observed prevalence in your sample

B.2 Probability

Symbol	Meaning
$P(A)$	Probability of event A
$P(A \mid B)$	Probability of A given B (conditional probability)
$P(A \cap B)$	Probability of both A and B
$P(A \cup B)$	Probability of A or B (or both)

B.3 Distributions

Notation	Distribution	Parameters
$X \sim N(\mu, \sigma^2)$	Normal	Mean $\mu$, variance $\sigma^2$
$X \sim \text{Bin}(n, p)$	Binomial	Trials $n$, success probability $p$
$X \sim \text{Pois}(\lambda)$	Poisson	Rate $\lambda$
$X \sim \text{Beta}(\alpha, \beta)$	Beta	Shape parameters $\alpha$, $\beta$

B.4 Regression

Symbol	Meaning
$y_i$	Outcome for patient $i$
$x_i$	Predictor value for patient $i$
$\beta_0$	Intercept
$\beta_1, \beta_2, \ldots$	Regression coefficients
$\hat{y}_i$	Predicted value for patient $i$
$\epsilon_i$	Residual (error) for patient $i$
$\hat{\beta}$	Estimated coefficient

Linear regression: $y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \epsilon_i$

Logistic regression: $\log\left(\frac{p_i}{1 - p_i}\right) = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots$

B.5 Penalised regression

Symbol	Meaning
$\lambda$	Penalty strength (regularisation parameter)
$\alpha$	Elastic net mixing parameter (0 = ridge, 1 = LASSO)
$\\|\boldsymbol{\beta}\\|_1 = \sum \|\beta_j\|$	L1 norm (sum of absolute values)
$\\|\boldsymbol{\beta}\\|_2^2 = \sum \beta_j^2$	L2 norm squared (sum of squares)

B.6 Survival analysis

Symbol	Meaning
$T$	Survival time (random variable)
$t$	A specific time point
$S(t) = P(T > t)$	Survival function
$h(t)$	Hazard function (instantaneous risk)
$H(t)$	Cumulative hazard
$\text{HR}$	Hazard ratio (from Cox model)

Cox model: $h(t \mid \mathbf{x}) = h_0(t) \exp(\beta_1 x_1 + \beta_2 x_2 + \cdots)$

B.7 Bayesian statistics

Symbol	Meaning
$\theta$	Parameter of interest
$P(\theta)$ or $\pi(\theta)$	Prior distribution
$P(D \mid \theta)$ or $L(\theta \mid D)$	Likelihood
$P(\theta \mid D)$	Posterior distribution
$P(D)$	Marginal likelihood (evidence)

Bayes’ theorem: $P(\theta \mid D) = \frac{P(D \mid \theta) \, P(\theta)}{P(D)}$

Or in words: Posterior $\propto$ Likelihood $\times$ Prior

B.8 Model performance

Symbol	Meaning
TP, FP, TN, FN	True/false positives/negatives
Se = TP / (TP + FN)	Sensitivity (recall)
Sp = TN / (TN + FP)	Specificity
PPV = TP / (TP + FP)	Positive predictive value (precision)
NPV = TN / (TN + FN)	Negative predictive value
AUC	Area under the ROC curve
NB	Net benefit (from decision curve analysis)

B.9 Subscripts and superscripts

Notation	Meaning
$x_i$	Value for individual $i$
$x_{ij}$	Value of variable $j$ for individual $i$
$\hat{\theta}$	Estimated value of $\theta$
$\theta^*$	Bootstrap replicate of $\theta$
$x^2$	$x$ squared
$\sqrt{x}$	Square root of $x$

B.10 Greek letters commonly used in statistics

Letter	Name	Common use
$\alpha$	Alpha	Significance level, elastic net mixing
$\beta$	Beta	Regression coefficients, Type II error rate
$\gamma$	Gamma	Effect modifier, shape parameter
$\delta$	Delta	Difference, effect size
$\epsilon$	Epsilon	Error term, small quantity
$\lambda$	Lambda	Rate parameter, penalty parameter
$\mu$	Mu	Population mean
$\pi$	Pi	Proportion, prior
$\sigma$	Sigma	Standard deviation
$\tau$	Tau	Between-study variance (meta-analysis)
$\chi^2$	Chi-squared	Chi-squared test statistic

--- title: "Mathematical Notation Reference" --- # Mathematical Notation Reference {#sec-math-notation .unnumbered} This appendix provides a quick reference for mathematical notation used throughout the course. You do not need to memorise these --- use this page as a lookup when you encounter unfamiliar symbols. ## Basic notation | Symbol | Meaning | Example | |--------|---------|---------| | $x$ | A variable (lowercase) | Blood pressure measurement | | $X$ | A random variable (uppercase) | Blood pressure in the population | | $\bar{x}$ | Sample mean | Mean blood pressure in your study | | $\mu$ | Population mean | True mean blood pressure | | $s$ or $\hat{\sigma}$ | Sample standard deviation | SD of blood pressures in your data | | $\sigma$ | Population standard deviation | True SD of blood pressures | | $n$ | Sample size | Number of patients | | $p$ | Probability or proportion | Prevalence of a disease | | $\hat{p}$ | Estimated proportion | Observed prevalence in your sample | ## Probability | Symbol | Meaning | |--------|---------| | $P(A)$ | Probability of event A | | $P(A \mid B)$ | Probability of A given B (conditional probability) | | $P(A \cap B)$ | Probability of both A and B | | $P(A \cup B)$ | Probability of A or B (or both) | ## Distributions | Notation | Distribution | Parameters | |----------|-------------|------------| | $X \sim N(\mu, \sigma^2)$ | Normal | Mean $\mu$, variance $\sigma^2$ | | $X \sim \text{Bin}(n, p)$ | Binomial | Trials $n$, success probability $p$ | | $X \sim \text{Pois}(\lambda)$ | Poisson | Rate $\lambda$ | | $X \sim \text{Beta}(\alpha, \beta)$ | Beta | Shape parameters $\alpha$, $\beta$ | ## Regression | Symbol | Meaning | |--------|---------| | $y_i$ | Outcome for patient $i$ | | $x_i$ | Predictor value for patient $i$ | | $\beta_0$ | Intercept | | $\beta_1, \beta_2, \ldots$ | Regression coefficients | | $\hat{y}_i$ | Predicted value for patient $i$ | | $\epsilon_i$ | Residual (error) for patient $i$ | | $\hat{\beta}$ | Estimated coefficient | **Linear regression:** $y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots + \epsilon_i$ **Logistic regression:** $\log\left(\frac{p_i}{1 - p_i}\right) = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \cdots$ ## Penalised regression | Symbol | Meaning | |--------|---------| | $\lambda$ | Penalty strength (regularisation parameter) | | $\alpha$ | Elastic net mixing parameter (0 = ridge, 1 = LASSO) | | $\|\boldsymbol{\beta}\|_1 = \sum |\beta_j|$ | L1 norm (sum of absolute values) | | $\|\boldsymbol{\beta}\|_2^2 = \sum \beta_j^2$ | L2 norm squared (sum of squares) | ## Survival analysis | Symbol | Meaning | |--------|---------| | $T$ | Survival time (random variable) | | $t$ | A specific time point | | $S(t) = P(T > t)$ | Survival function | | $h(t)$ | Hazard function (instantaneous risk) | | $H(t)$ | Cumulative hazard | | $\text{HR}$ | Hazard ratio (from Cox model) | **Cox model:** $h(t \mid \mathbf{x}) = h_0(t) \exp(\beta_1 x_1 + \beta_2 x_2 + \cdots)$ ## Bayesian statistics | Symbol | Meaning | |--------|---------| | $\theta$ | Parameter of interest | | $P(\theta)$ or $\pi(\theta)$ | Prior distribution | | $P(D \mid \theta)$ or $L(\theta \mid D)$ | Likelihood | | $P(\theta \mid D)$ | Posterior distribution | | $P(D)$ | Marginal likelihood (evidence) | **Bayes' theorem:** $P(\theta \mid D) = \frac{P(D \mid \theta) \, P(\theta)}{P(D)}$ Or in words: Posterior $\propto$ Likelihood $\times$ Prior ## Model performance | Symbol | Meaning | |--------|---------| | TP, FP, TN, FN | True/false positives/negatives | | Se = TP / (TP + FN) | Sensitivity (recall) | | Sp = TN / (TN + FP) | Specificity | | PPV = TP / (TP + FP) | Positive predictive value (precision) | | NPV = TN / (TN + FN) | Negative predictive value | | AUC | Area under the ROC curve | | NB | Net benefit (from decision curve analysis) | ## Subscripts and superscripts | Notation | Meaning | |----------|---------| | $x_i$ | Value for individual $i$ | | $x_{ij}$ | Value of variable $j$ for individual $i$ | | $\hat{\theta}$ | Estimated value of $\theta$ | | $\theta^*$ | Bootstrap replicate of $\theta$ | | $x^2$ | $x$ squared | | $\sqrt{x}$ | Square root of $x$ | ## Greek letters commonly used in statistics | Letter | Name | Common use | |--------|------|-----------| | $\alpha$ | Alpha | Significance level, elastic net mixing | | $\beta$ | Beta | Regression coefficients, Type II error rate | | $\gamma$ | Gamma | Effect modifier, shape parameter | | $\delta$ | Delta | Difference, effect size | | $\epsilon$ | Epsilon | Error term, small quantity | | $\lambda$ | Lambda | Rate parameter, penalty parameter | | $\mu$ | Mu | Population mean | | $\pi$ | Pi | Proportion, prior | | $\sigma$ | Sigma | Standard deviation | | $\tau$ | Tau | Between-study variance (meta-analysis) | | $\chi^2$ | Chi-squared | Chi-squared test statistic |

Notation	Distribution	Parameters
\(X \sim N(\mu, \sigma^2)\)	Normal	Mean \(\mu\), variance \(\sigma^2\)
\(X \sim \text{Bin}(n, p)\)	Binomial	Trials \(n\), success probability \(p\)
\(X \sim \text{Pois}(\lambda)\)	Poisson	Rate \(\lambda\)
\(X \sim \text{Beta}(\alpha, \beta)\)	Beta	Shape parameters \(\alpha\), \(\beta\)

Symbol	Meaning
\(y_i\)	Outcome for patient \(i\)
\(x_i\)	Predictor value for patient \(i\)
\(\beta_0\)	Intercept
\(\beta_1, \beta_2, \ldots\)	Regression coefficients
\(\hat{y}_i\)	Predicted value for patient \(i\)
\(\epsilon_i\)	Residual (error) for patient \(i\)
\(\hat{\beta}\)	Estimated coefficient

Notation	Meaning
\(x_i\)	Value for individual \(i\)
\(x_{ij}\)	Value of variable \(j\) for individual \(i\)
\(\hat{\theta}\)	Estimated value of \(\theta\)
\(\theta^*\)	Bootstrap replicate of \(\theta\)
\(x^2\)	\(x\) squared
\(\sqrt{x}\)	Square root of \(x\)