density, fraction, the distance of two elements in the interval

Question

Discretize the interval $[w_l, w_h]$ into $n$ elements. Let $f_n(w)$ be the fraction of $w$ with $n$ elements in the interval. I saw the formula in some paper that $$f(w) = \lim_{n \to \infty} \frac{f_n(w)}{\Delta w}$$ where $f$ is density and $\Delta w = \frac{w_h - w_l}{n}$. This might be the basic question, but I want to understand the intuition behind this equation. Why is the limit of the ratio of the fraction to the interval gap interpreted as density?

Answer 1

A probability density of a random variable $W$ at $w$ (i.e., $f_W(w)$) is the rate at which you accumulate probability at that point:

$$f_W(w) := \frac{d}{dw}P(W\leq w)$$

In your example, $W \in [w_l,w_h] \subset \mathbb{R}$.

Let's say that for any $n \in \mathbb{N}$ we can calculate $$p_n(i):=P\left(W \in I_{n,i}\right) \;\forall i \in \{1,...,n\}$$

Where

$I_{n,i} := [w_l+ (i-1)\Delta_n w, w_l+i\Delta_n w]$ and

$\Delta_n w_:= \frac{w_h - w_l}{n}$

Also, given any $w \in [w_l,w_h]$ we can find which $I_i$it is in:

$$i_n(w) = \sum_{k=1}^nk\mathbb{1}_{I_{n,k}}(w) $$

Putting this together, we see that what you call $f_n(w)$ is more precisely defined as:

$$f_n(w):= \frac{p_n(i_n(w))}{\Delta_n w}$$

Put simply this is the average value of $f_W$ on the interval $I_{n,i}$.

Given that and if we assume that $P(W\leq w)$ is differntiable, then you can see we have $\lim_{n \to \infty} \Delta_n w = 0$ and $f_n(w) \to \frac{d}{dw}P(W\leq w)$ (by the mean value theorem).