Appendix B
Symbolism
In part, we use different notation to other mathematical disciplines. We do not use capital letters as in probability theory to denote random variables but denote them by bold printing. We do this not only to distinguish between a random variable F with F‐distribution and its realisation F but mainly because linear models are important in this book. In a mixed model in the two‐way cross classification of the analysis of variance with a fixed factor A and a random factor B is the model equation with capital letters written as:
This looks strange and is unusual. We use instead
Functions are never written without an argument to avoid confusion. So p(y) is often a probability function but p a probability. Further is f(y) a density function but f the symbol for degrees of freedom.
| Sense | Symbol |
| Rounding up function | ⌈x⌉ = smallest integer ≥x |
| Binomial distribution with parameters n, p | B(n,p) |
| Chi‐squared (χ2)‐distribution with f degrees of freedom | CS (f) |
| Determinant of the matrix A | |A|, det(A) |
| Diagonal matrix of order n | D(a1. … . an) |
| Direct sum of the sets A and B | A ⊕ B |
| Identity matrix of order n | In |
| (n x m)‐matrix with only zeros | 0nm |
| (n x m)‐matrix with only ones | 1nm |
| Euclidian space of dimension n and 1 respectively (real axis), positive real axis | Rn; R1 = R; R+ |
| y is distributed as | y∼ |
| Sense | Symbol |
| Indicator function | Is A a set and x ∈ A, then  |
| Kronecker delta |  |
| Interval on the x‐axis | |
|
Open Half‐open Closed |
(a, b) : a < x < b [a, b): a ≤ x < b (a, b]: a < x ≤ b [a, b] : a ≤ x ≤ b |
| ith order statistic | y(i) |
| Cardinality (number) of elements in S | card (S); |S| |
| Constant in formulae | const |
| Empty set | Ø |
| Multivariate normal distribution with expectation vector μ and covariance matrix Σ | N(μ,Σ) |
| Normal distribution with expectation μ and variance σ2 | N(μ,σ2) |
| Null vector with n elements | 0n |
| Vector with n ones | 1n |
| Parameter space | Ω |
| Poisson distribution with parameter λ | P(λ) |
| P‐quantile of the N(0, 1)‐distribution | z(P) or zP |
| P‐quantile of the χ2‐distribution with f degrees of freedom | CS(f, P ) |
| P‐quantile of the t‐distribution with f degrees of freedom | t(f, P) |
| P‐quantile of the F‐distribution with f1 and f2 degrees of freedom | F(f1, f2, P) = FP(f1, f2) |
| Rank of matrix A | rk(A) |
| Rank space of matrix A | R(A) |
| Standard normal distribution with expectation 0; variance 1 | N(0,1) |
| Trace of matrix A | tr(A) |
| Transposed vector of Y | YT |
| Vector (column vector) | Y |
| Distribution function of a N(0,1)‐distribution | Φ(x) |
| Distribution Function | F(y) = P(y ≤ y) |
| Density function of a N(0,1)‐distribution | φ(x) |
| Random variable (bold print) | y, Y |