Approximating Discrete Random Variables as Continuous

When a discrete random variable takes many closely-spaced values, we can replace its discrete distribution with a continuous approximating density.

Suppose $X$ takes values $x_1 < x_2 < \cdots < x_N$ evenly spaced by $\Delta x.$ If $\Delta x$ is small, we match each discrete probability to a density $f$ via

$$P(X = x_i) \approx f(x_i) \cdot \Delta x,$$

so that

$$f(x_i) \approx \dfrac{P(X = x_i)}{\Delta x}.$$

The function $f$ then plays the role of a probability density for $X$ on the continuous range.

Example. Let $X$ take values $x_k = k/N$ for $k = 1, 2, \ldots, N$ with $P(X = x_k) = 1/N$ (uniform on these values). The spacing is $\Delta x = 1/N,$ so

$$f(x_k) \approx \dfrac{1/N}{1/N} = 1.$$

The approximating density is $f(x) = 1$ on $[0, 1]$ — the continuous uniform distribution on $[0,1].$