Appendix F: Notation Guide

This appendix collects all mathematical notation used in the book, organized by topic. Where a symbol has multiple meanings depending on context, each meaning is listed with the chapters in which it appears.

F.1 Probability and Statistics Notation

Symbol Meaning Chapters
Omega Sample space 1, 2
A, B, C Events 1, 2
A^c Complement of event A 2
A intersection B Intersection of events A and B 2
A union B Union of events A and B 2
P(A) Probability of event A 1-42
P(A|B) Conditional probability of A given B 2, 9, 25
E[X] Expected value of random variable X 1-42
Var(X) Variance of X 3, 6, 8
SD(X) Standard deviation of X 3, 6, 8
Cov(X, Y) Covariance of X and Y 6, 15
Corr(X, Y) or rho Pearson correlation coefficient 6, 15, 29
X ~ D X is distributed according to distribution D 3, 7, 9
mu Population mean 3, 5
sigma Population standard deviation 3, 5, 6
sigma^2 Population variance 3, 5
x-bar Sample mean 3, 5
s Sample standard deviation 3, 5
s^2 Sample variance 3, 5
p Probability parameter (Bernoulli/Binomial) 1-42
p-hat Sample proportion 4, 5
n Sample size 3-42
N(mu, sigma^2) Normal distribution with mean mu, variance sigma^2 3, 5, 6
Binom(n, p) Binomial distribution 3, 4, 8
Pois(lambda) Poisson distribution with rate lambda 7, 33, 35
Beta(a, b) Beta distribution with shape parameters a, b 9
Gamma(alpha, beta) Gamma distribution 25
Exp(lambda) Exponential distribution with rate lambda 25
t_nu Student's t-distribution with nu degrees of freedom 5
chi^2_k Chi-squared distribution with k degrees of freedom 5, 17
F_{d1, d2} F-distribution with d1, d2 degrees of freedom 16
Phi(z) Standard normal CDF 6, 7
phi(z) Standard normal PDF 6
z_alpha z-critical value for significance level alpha 5
H_0 Null hypothesis 5
H_a or H_1 Alternative hypothesis 5
alpha Significance level / Type I error rate 5
beta (stats) Type II error rate 5
1 - beta Statistical power 5
p-value Probability of data at least as extreme under H_0 5
CI Confidence interval 5
df Degrees of freedom 5
R^2 Coefficient of determination 6, 13
R^2_adj Adjusted R-squared 6, 13
AIC Akaike Information Criterion 12
BIC Bayesian Information Criterion 12
L(theta) Likelihood function 12
ell(theta) Log-likelihood function 12
MLE Maximum likelihood estimate 12
1_A or I(A) Indicator function (1 if A true, 0 otherwise) 2, 12

F.2 Betting-Specific Notation

Symbol Meaning Chapters
d Decimal odds 1, 8, 10
b Net payout per unit staked (b = d - 1) 8
a American odds 1
p_imp Implied probability (1/d) 1, 10
p_fair Fair (no-vig) probability 1, 10
p_true True probability estimated by model 1-42
v Vigorish (overround) 1, 10
f Fraction of bankroll wagered 8
f* Optimal Kelly fraction 8
B_0 Initial bankroll 8
B_n Bankroll after n bets 8
G(f) Expected log-growth rate as function of f 8
EV Expected value of a bet 1, 8
ROI Return on investment (profit / total staked) 1, 10, 38
CLV Closing line value 10, 38
ATS Against the spread 6, 20
S Point spread 6, 20, 21
T Total (over/under line) 7, 20, 21
M Margin of victory (actual score difference) 6, 20
O Overround (sum of implied probabilities minus 1) 1, 10
edge Estimated edge: p_true * d - 1 1, 8, 10
yield Profit per unit staked (expressed as percentage) 38
W Number of winning bets 3, 4, 8
L Number of losing bets 3, 4, 8
SR Strike rate (win rate) 3, 4
MDD Maximum drawdown 8, 38
Sharpe Sharpe ratio (risk-adjusted return) 38

F.3 Linear Algebra Notation

Symbol Meaning Chapters
x (bold lowercase) Column vector 13-16, 18
X (bold uppercase) Matrix (typically design/feature matrix) 13-16
x_i i-th element of vector x 13-16
X_{ij} Element in row i, column j of X 13-16
X^T Transpose of matrix X 13-16
X^{-1} Inverse of square matrix X 13
I or I_n n x n identity matrix 13
0 Zero vector or zero matrix 13
det(A) Determinant of A 13
tr(A) Trace of A (sum of diagonal elements) 15
||x|| Euclidean norm of x 13, 14
||x||_1 L1 norm (sum of absolute values) 13
||x||_2 L2 norm (Euclidean norm) 13, 14
x . y Dot product of x and y 13, 14
diag(d) Diagonal matrix with vector d on diagonal 15
rank(A) Rank of matrix A 15
lambda_i i-th eigenvalue 15
v_i i-th eigenvector 15

F.4 Calculus Notation

Symbol Meaning Chapters
f'(x) or df/dx Derivative of f with respect to x 8, 12
f''(x) or d^2f/dx^2 Second derivative 8
partial f / partial x_i Partial derivative of f with respect to x_i 12, 14
nabla f or grad f Gradient vector of f 14, 18
H or nabla^2 f Hessian matrix (matrix of second partial derivatives) 14
integral f(x) dx Indefinite integral A.5
integral_a^b f(x) dx Definite integral from a to b A.5, 9
sum_{i=1}^{n} Summation from i=1 to n Throughout
prod_{i=1}^{n} Product from i=1 to n 8, 12
lim_{n->inf} Limit as n approaches infinity 8
argmax_x f(x) Value of x that maximizes f 12, 14
argmin_x f(x) Value of x that minimizes f 14
O(n) Big-O notation: asymptotic upper bound 14

F.5 Machine Learning Notation

Symbol Meaning Chapters
w Weight vector 13-18
w_0 or b Bias / intercept term 13-18
eta Learning rate 14
lambda Regularization strength 13, 14
J(w) or L(w) Loss / cost function 14
y True label / target variable 13-18
y-hat Predicted value 13-18
sigma(z) Sigmoid function: 1/(1 + e^{-z}) 13, 14
softmax(z)_i exp(z_i) / sum_j exp(z_j) 14, 18
ReLU(z) max(0, z) 14
D_train Training dataset 13-18
D_val Validation dataset 13-18
D_test Test dataset 13-18
m Number of training examples 14
k Number of features (or classes, depending on context) 13-18
epoch One complete pass through the training data 14
batch Subset of training data used in one gradient step 14
theta General model parameter vector 12-18
p(y|x; theta) Probability of y given x under parameters theta 12, 13
CE(y, y-hat) Cross-entropy loss 14, 17
MSE Mean squared error 13
MAE Mean absolute error 13
RMSE Root mean squared error 13
AUC Area under the ROC curve 17
AP Average precision 17
BS Brier score 17
F1 F1 score (harmonic mean of precision and recall) 17
TP, FP, TN, FN True/false positives/negatives 17
CV Cross-validation 16
k-fold k-fold cross-validation 16
h^{(l)} Hidden layer activations at layer l 14, 18
W^{(l)} Weight matrix at layer l 14, 18
T Number of trees (in ensemble methods) 16
depth Maximum tree depth 16
alpha (ML) Mixture or attention weight 18
dropout(p) Dropout with probability p 14, 18

F.6 Sport-Specific Notation

Symbol Meaning Sport Chapters
R_A, R_B Elo ratings of teams A and B All 9, 20-24
K Elo K-factor (update magnitude) All 9
HFA Home field/court advantage (in points or rating) All 6, 9, 20-24
lambda_H, lambda_A Poisson scoring rates (home, away) Soccer, Hockey 7, 33, 35
xG Expected goals Soccer 33
EPA Expected points added NFL 20, 21
WPA Win probability added NFL, MLB 20, 25
DVOA Defense-adjusted Value Over Average NFL 20
PER Player Efficiency Rating NBA 22
WAR Wins Above Replacement MLB, NBA 23, 24
RAPTOR Robust Algorithm using Player Tracking and On/Off Ratings NBA 22
FIP Fielding Independent Pitching MLB 23
Corsi Shot attempt differential (for/against) NHL 24
PDO Shooting% + Save% (regression indicator) NHL 24
ORtg, DRtg Offensive/Defensive rating (points per 100 possessions) NBA 22
Pace Possessions per game NBA 22
BABIP Batting Average on Balls In Play MLB 23

F.7 Conventions and Formatting

  • Boldface denotes vectors and matrices: x, W, X.
  • Italics denote scalar variables: x, p, lambda.
  • A hat (^) over a symbol denotes an estimator: theta-hat, y-hat, p-hat.
  • A tilde (~) means "distributed as": X ~ N(0, 1).
  • A bar over a symbol denotes a sample mean: x-bar = (1/n) * sum x_i.
  • Subscripts generally index elements (x_i) or time steps (B_t).
  • Superscripts in parentheses index layers in neural networks: W^{(l)}.
  • When a symbol serves double duty, the context makes the intended meaning clear. For example, alpha is the significance level in hypothesis testing (Chapter 5) and the learning rate or mixture weight in some machine learning contexts (Chapter 18). Similarly, beta denotes both regression coefficients and Type II error rates.

Cross-reference this guide with Appendix A (Mathematical Foundations) for definitions and Appendix E (Glossary) for plain-language descriptions of terms.