What is the difference between parameter and variable? This is a question that I have in order to reconcile a difference in terminology.
In the linear regression setting, we have $y=\beta x + \theta$. Here, we call $x$ a variable. When we are trying to use this in a practical setting, suppose $x \in \mathbb{R}^n$. Then $x$ is represented as a feature vector. I think each feature could also be called a parameter. Is that true?
But if we can indeed call these parameters, then, since each $x_i$ is an iid variable, does this mean that these parameters are parameters of the distribution from which the $x_i$ are drawn? They can certainly be used to determine the parameters of the distribution.
 A: Conventionally, "parameter" refers to the numbers that govern the statistical model or structure of data. Parameters are normally structural and descriptive. So in your case $\beta$ is a parameter, but $x$ is "data" and not a "parameter".  If $x$ is drawn from a random distribution as in your comment, $\mu$ and $\sigma$ would also be parameters.  "Feature" is machine learning lingo for a component of the data.  In your case you just have one feature, since it's one-dimensional
A: In statistics, $x$ is always called a variable, Beta and Theta are always parameters.  In other fields, $x$ may be called a parameter.  We can't help that but at least in statistics we are consistent (I hope).
A: The way I've always looked at these semantics is the following:
When collecting data a 'variable' is something you measure (in my field of biomedical research this could be sex, height, disease severity, diagnosis groups, etc.).
In statistics, however, these variables are often 'dissected' into vectors of data which are ready for analysis. For linear regression for example, all your variables should be reclassified so that each parameter of your model represents one degree of freedom and receives its own coefficient. So in the case of continuous variables nothing changes (the coefficient represents the change in outcome when the continuous variable goes up or down one unit). For categorical variables with more than 2 levels however, you'll have to create dummy data vectors for all levels except the reference levels. This way, a four level categorical variable will be represented in your regression a three parameters (one for all levels except the reference), and each parameter/level will be assigned a coefficient during model fitting. 
Compare these models to predict someone's weight
$x$ = length, in a one variable model and one parameter model to predict weight ($y$):
$y = a + Bx + e$ with
$x$ = US state in a one variable model, 50 parameter model to predict weight ($y$), with New York state as reference:
$y = a + B1x1 + ... + B50x50 + e$
where all states except NY are recoded as dummy parameters x1-x50 (including D.C. ofcourse)
In short, IMO in the context of regression a 'variable' constitutes the raw data, while a parameter reflects the actual $x$'s in your model.
A: Variable
In the field of mathematics, a variable defines as an element connected to a number, known as an estimation of the variable that is self-estimated, not completely determined, or ambiguous. The expression “variable” originates from the way that, when the argument (additionally called the “variable of the Function”) changes, then the estimate changes accordingly.
Parameter
A parameter, by and large, is an entity that can help in connecting or grouping a specific framework. That is, a parameter is a component of a system that is helpful, or basic. Inside and over different fields, watchful refinement must be kept up of the diverse utilizations of the term parameter and of different terms frequently connected with it.
The reference of this site is https://researchpedia.info/difference-between-variable-and-parameter/
