# A brief primer on scientific and mathematical notations

As I finished writing the final draft of my first first author paper, survClust, there were a lot of other firsts! In my opinion writing the methods and a crisp conclusion and discussion were the difficult parts.

Below, I share my notes that really came in handy while I was writing the methods section of my manuscript.

## 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 -

$Pr[a\leq X\leq b]=\int _{a}^{b}f_{X}(x)\,dx$

Hence, if $$F_{X}$$ is the cumulative distribution function of $$X$$ then:

$F_{X}(x)=\int _{-\infty }^{x}f_{X}(u)\,du,$

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., $$\widehat {\theta }$$ is an estimator for $$\theta$$ .

• Building on the above point the sample mean, variance and correlation coefficient are denoted as $$\bar{x}, s^2, r$$ 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 $$\textbf{X}$$, 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 $$\textbf{X}^T = (X_1,X_2,...,X_p)$$, we predict the output $$\textbf{Y}$$ via a simple linear regression -

$\hat{\textbf{Y}} = \hat\beta_0 + \sum_{n=1}^{p} \textbf{X}_{j}\hat\beta_{j}$ Or writing this in a vector form as an inner product - $$\hat{\textbf{Y}} = \textbf{X}^T\hat\beta$$ 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,

$RSS(\beta) = (\textbf{y} - \textbf{X}\beta)^T(\textbf{y} - \textbf{X}\beta)$ where $$\textbf{X}$$ is an $$N × p$$ matrix with each row an input vector, and $$\textbf{y}$$ is an $$N$$-vector of the outputs. See how $$\textbf{y}$$ 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!

##### Arshi Arora
###### Research Biostatistician

New Yorker from Jaipur, India. Cancer Genomics, pottery and biking.