Cheat Sheet

Mathematical Notations

• Scalar: Non-bold lower case letter. Example $y$
• Column vector: Bold lower case letter. Example $\boldsymbol{x}$
• Matrix: Upper case letter. Example $A$
• A set of scalars/vectors: Curly brackets. Example $\left\lbrace2, 3, 5, 7\right\rbrace$, $\left\lbrace y_i\right\rbrace$ or $\left\lbrace\boldsymbol{x}_i\right\rbrace$
• An element of a vector, a matrix or a set: Subscript index. Example $x_i$ or $a_{i,j}$
• The value in an iterative process: Superscript index in parentheses. Example $\theta^{\left(t\right)}$

Probability Notations

• A single random sample: $\omega$
• A random variable (RV): $X\left(\omega\right)$ (usually shorten to $X$. Note that capital letters are also used for matrices)
• A realization of an RV: $x=X\left(\omega\right)$
• The probability of an event $Pr\left(0.2<X<0.6\right)$
• Probability Mass Function (PMF): (for a discrete RV) $p_X\left(x\right)=Pr\left(X=x\right)$
• Cumulative Distribution Function (CDF): $F_X\left(x\right)=Pr\left(X\leq x\right)$
• Probability Density Function (PDF): $f_X\left(x\right)$ define by $F_X\left(x\right)=\int_{-\infty}^xf_Xdx$

Common notation to describe a dataset:

• $N$ - The number of samples in a dataset.
• $d$ The dimension of the data.
• $\boldsymbol{x}$ - A vector of measured values related to a single sample.
• $\left\lbrace\boldsymbol{x}_i\right\rbrace$ - The set of measurements in a dataset.
• $X$ - The matrix of measurements $\left[\boldsymbol{x}_0,\boldsymbol{x}_1,\ldots,\boldsymbol{x}_{N-1}\right]^T$.

(Note that $X$ is also sometime used as the random variable generating $\boldsymbol{x}$).

• $x_{i,j}$ - The $j$-th element of the $i$-th measurement (which is the $\left(i,j\right)$-th element of the matrix $X$).
• $\left\lbrace y_i\right\rbrace$ - The set of labels in a dataset.
• $\boldsymbol{y}$ - the vector of labels $\left[y_0,y_1,\ldots,y_{N-1}\right]^T$.
• $y_i$ - The $i$-th label (which is the $i$-th element of the vector $\boldsymbol{y}$).