How many ways to represent a probability density function?

Question

I have read accidentally in a book this sentence: " ... consider a random sample $X_1, X_2, \ldots, X_n$, each $X_i$ having probability distribution $f(x)dx$. Thus, we have $$\mathbb{P}(X_1\in dx_1,\ldots, X_n\in dx_n)=\Pi_{i=1}^n f(x_i)dx_i$$

and where $\ldots$"

I guess that $f(x)dx$ is the probability density function of random variable $X$ but it seems strange to what I have learnt in my probability course. Is there some different ways to represent a probability denstiy function? Does anyone can explain to me? Is there some books where I can find these definition about it? Thanks in advance.

Answer 1

usually pdfs are denoted just $f(x)$. For continuous rvs I guess the notation you have means 'in the arbitrarily small neighborhood of $x_1$. A good way to look at pdfs is the derivative of cdfs. If cdf of $X$ is $P(X <x)$, then pdf is $$ \lim_{\epsilon \to 0} \frac{P(X <x +\epsilon) - P(X <x)}{\epsilon} = f(x) $$

Answer 2

I think that this is an alternative definition. Following the answer of Alex, if $dx$ is an infinitely small number and $f$ is assumed to be continuous, the probability that $X$ is included within the interval $(x,x+dx)$ is equal to

$$\mathbb{P}\left(x<X<x+dx\right) = f(x)dx$$

Or you can find an explaination here: http://mathinsight.org/probability_density_function_idea and there: http://www.eecs.berkeley.edu/~dtse/cs70_fall09/note18_fall09.pdf