I have been given a question on the following pdf:
Suppose the random variable, X, follows a uniform distribution on the interval (0, θ). The pdf of X is $f(x;θ)$ = $1/θ$, $if$ $0≤x≤θ$, $θ>0$, $0$ $otherwise$
I would really like some simplified explanation of the notation here, my confusion comes in a couple of places, a) is the semicolon just alternative to using a comma and is therefore just separating variables of a function, b) and if so why is x defined and it's domain specified but not used in the function?
I was asked to find the cdf of the above pdf, which I am aware of how to do normally but the notation has confused me.
***Note: If it helps the questions on the pdf are related to likelihood functions.
The semicolon separates variables from parameters. Think of it as there being a set of pdfs, each of which is defined in terms of a variable $x$, but which differ from each other in their parameter $\theta$. By setting the parameter you are basically picking out one specific pdf of all these possible pdfs.
A function does not necessarily need to use its variables. A typical example is a constant function such as $g(x) = 2$ defined on the real numbers. Whatever real number you put into the function $g$, you always get the same result back, namely 2. Your case is similar. Whatever value $0 \le x \le \theta$ you put into the function, you always get $1/\theta$ back.