A brief primer on scientific and mathematical notations

What this is?

Notes on how to describe a statistical methodology. Some basic rules and notations that you should keep in mind.

Scientific notations

• Random variables are usually written in uppercase roman letters: $X,Y$, etc.

• Probability density functions (pdfs) and probability mass functions are denoted by lowercase letters, e.g. $f_{(x)}$, or $f_X(x)$.

• Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. $F(x)$, or $F_X(x)$.

Let's summarize the above three points with an example -

A random variable $X$ has density $f_X$ as follows -

$$Pr[a\le X\le b]={\int }_a^b{f}_X(x)dx$$

Hence, if $F_X$ is the cumulative distribution function of $X$ then:

$$F_X(x)={\int }_{-\mathrm{\infty }}^x{f}_X(u)du,$$

and

$$f_X(x)=\frac{d}{dx}{F}_X(x).$$

Now, let's go over some quick statistical nitty-gritties:

• Greek letters $\theta ,\beta$ are commonly used to denote unknown parameters.

• Placing a hat, or caret, over a true parameter denotes an estimator of it, e.g., $\hat{\theta }$ is an estimator for $\theta$ .

• Building on the above point the sample mean, variance and correlation coefficient are denoted as $\overline{x},s^2,r$ respectively. On the other hand population parameters are represented as follows - population mean $\mu$, population variance ${\sigma }^2$, and population correlation as $\rho$

Finally most of the time you will need to know the following writing notions while drafting the methods section of your manuscript -

• Input or independent variables are denoted by $X$, output or dependent variables are denoted by $Y$, and qualitative outputs by $G$.

• If $X$ is a vector, annotate its values by subscripts $X_j$

• Observed values are written in lowercase; hence the $i^{th}$ observed value of $X$ is written as $x_i$, where $x_i$ is a scalar or vector.

• Matrices are represented by bold uppercase letters; for example a matrix $\mathbf{\text{X}}$, with dimensions $N$ x $p$ i.e a set of $N$ input $p$-vectors. In general, vectors will not be bold, except when they have $N$ components; Note that all vectors are assumed to be column vectors.

Let's break it down with an example -

Given a vector of inputs ${\mathbf{\text{X}}}^T=(X_1,X_2,...,X_p)$, we predict the output $\mathbf{\text{Y}}$ via a simple linear regression -

$$\widehat{\mathbf{\text{Y}}}={\hat{\beta }}_0+\sum _{n=1}^p{\mathbf{\text{X}}}_j{\hat{\beta }}_j$$ Or writing this in a vector form as an inner product - $\widehat{\mathbf{\text{Y}}}={\mathbf{\text{X}}}^T\hat{\beta }$ To solve this we need to estimate a value of $\beta$ such that it minimizes the Residual Sum of Squares or RSS as follows -

$$RSS(\beta )=\sum _{i=1}^N(y_i-x_i^T\beta {)}^2$$

Or in matrix notation we can write it as,

$$RSS(\beta )=(\mathbf{\text{y}}-\mathbf{\text{X}}\beta {)}^T(\mathbf{\text{y}}-\mathbf{\text{X}}\beta )$$ where $\mathbf{\text{X}}$ is an $N\times p$ matrix with each row an input vector, and $\mathbf{\text{y}}$ is an $N$-vector of the outputs. See how $\mathbf{\text{y}}$ is in bold in the above question.

Or take one of your favorite papers, and try to go over its methods section to iron and figure out other key details!