如何用无穷小从PMF推导PDF并构造性论证连续随机变量及PDF

Hey there! Let's tackle these two questions using infinitesimal reasoning—no formal ε-δ calculus needed, just straight-up constructive, intuitive logic that connects discrete PMFs to continuous PDFs. Here's how it works:

First, let's recap discrete random variables: a PMF ( p(x_k) = P(X = x_k) ) gives the exact probability that the variable takes on a specific discrete value ( x_k ).

To bridge this to continuous variables, think of a continuous random variable as a discrete variable where the "gap" between adjacent possible values shrinks to an infinitesimal ( dx ) (a tiny, tiny quantity that's effectively zero, but we can still do arithmetic with it).

For a discrete variable, if we group adjacent values into an infinitesimal interval ([x, x+dx]), the probability that the variable falls into this interval is roughly the PMF value at ( x ) (since the gap is infinitesimal, there's basically only one discrete value in the interval). We can rewrite this probability as a density times interval length:
[ P(x \leq X \leq x+dx) \approx f(x) \cdot dx ]
Here, ( f(x) ) is the PDF. Rearranging this gives us the direct connection to the PMF:
[ f(x) = \frac{P(X = x)}{dx} ]
In plain terms, the PDF is the "probability per unit infinitesimal length"—you're taking the discrete point probability (PMF) and spreading it over the infinitesimal interval to get a density measure.

We can build up continuous random variables and PDFs step-by-step from discrete ones using infinitesimals, like this:

• Start with a sequence of discrete variables: Define a set of discrete random variables ( {X_m} ), where each ( X_m ) takes values ( x = k \cdot \Delta x_m ) (for integer ( k )). Here, ( \Delta x_m ) is an infinitesimal that gets smaller as ( m ) increases (think of it as making the gaps between discrete points tinier and tinier). Each ( X_m ) has a PMF ( p_{m,k} = P(X_m = k \cdot \Delta x_m) ), with the usual discrete normalization: ( \sum_k p_{m,k} = 1 ).

• Define the PDF as a density ratio: For any point ( x ), pick the discrete value ( k \cdot \Delta x_m ) that's exactly ( x ). The probability that ( X_m ) falls into the infinitesimal interval ([x, x+\Delta x_m]) is just ( p_{m,k} ). We define the PDF ( f(x) ) as the ratio of this discrete probability to the interval length:
  [ f(x) = \frac{p_{m,k}}{\Delta x_m} ]
  This gives us a finite value (since ( p_{m,k} ) is also infinitesimal—total probability is 1, so more discrete points mean smaller individual probabilities—and the ratio of two infinitesimals can be finite).

• Verify the continuous normalization: For the discrete variable, ( \sum_k p_{m,k} = \sum_k f(k \cdot \Delta x_m) \cdot \Delta x_m = 1 ). As ( \Delta x_m ) becomes infinitely small, this sum turns into an integral (summing over infinitely many infinitesimal intervals):
  [ \int_{-\infty}^{\infty} f(x) dx = 1 ]
  This is exactly the normalization condition for a continuous PDF.

• Define continuous probabilities: For any interval ([a,b]), the probability that our continuous variable ( X ) falls into it is the sum of the probabilities of all infinitesimal subintervals within ([a,b]). Translating that from discrete sums to continuous integrals, we get:
  [ P(a \leq X \leq b) = \int_a^b f(x) dx ]

• Wrap up the continuous variable definition: The continuous random variable ( X ) is just the limit of the discrete sequence ( {X_m} ) as ( \Delta x_m ) shrinks to an infinitesimal. Instead of taking on exact discrete values, ( X ) can take any value in ( \mathbb{R} ), and its probabilities are defined by integrating the PDF we constructed from the PMF ratios.

内容的提问来源于stack exchange，提问作者user188529