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If $X \in \mathbb{R}$ is a continuous random variable then its density $f$ (if it exists) is defined as $$ f(x) = \lim_{\epsilon \rightarrow 0} \frac{F(x + \epsilon) - F(x)}{\epsilon}, $$ where $F(x) = \Pr(X \leq x)$. It is sometimes more convenient to think of it as $f(x) = \Pr(X = x)$. However, this cannot be written! Is it more rigourous (i.e., can it be written) to write $f(x) = \Pr(X \in I_x)$, where $I_x$ is a neightbourhood of $x$? |
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Do you have a particular probability space in mind where you want to do this calculation? For a random variable $X$ that admits a probability density function $f(x)$, you can think of $(X\in I_{x})$ as the event of being in the neighborhood $I_{x}$ of some specific point $x$. The then probability of that event is just $$ P(X\in I_{x}) = \int_{I_{x}}f(y)dy.$$ Do you want something more than this? For the one-dimensional case, I suppose you can think of the density as being the differential probability of being in an infinitesimal neighborhood that has the point $x$ as its left endpoint: $$ P(X\in [x,x+h)) = F(x+h) - F(x) = \int_{x}^{x+h}f(y)dy \approx f(x)\cdot{}h$$ for small $h$. Note that you can get the same thing by computing a Taylor series expansion of $F(x)$ at the point $x$, and then evaluating the resulting expansion at $x+h$, and considering only the linear approximation. |
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I think the closest you can get to the concept you want to express without risking confusion is $f(x)=\frac{\Pr(x<X \le x+dx)}{dx}$. This is a bit of an abuse of notation unless $\Pr(x<X \le x+dx)$ is defined to be equivalent to the differential $dF(x)$. |
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