What is probability-frequency function?

Question

What is probability-frequency function and how it differs from probability function?

Answer 1

If you observe a random sample $X_1,\ldots,X_n$ from some discrete density $f$ and wish to estimate that density, one way is to note the frequencies of each occurrence of elements from the sample space $S$:

$$ \hat{f}(x) = \frac{1}{n}\sum_{i=1}^n 1_{\{ X_i = x\}}. $$ This estimate converges a.s. to $f(x) = {\bf E}[1_{\{ X = x \}}]$. You are essentially building an empirical distribution, which can also be done in the continuous setting by looking at something like $$ \hat{F}(x) = \frac{1}{n}\sum_{i=1}^n 1_{\{ X_i \leq x\}}. $$ However, the term "frequency" doesn't make much sense in the continuous setting since the probability of observing any fixed value $x$ is $0$.