What does the notation $\pi$ mean in probability and statistics

Question

I have noticed that in the presentation of probability distribution, sometimes a certain probability is dentoed by $\pi(...)$. For example, in the wike page of Beta binomial distribution https://en.wikipedia.org/wiki/Beta-binomial_distribution, they use $P(X=k|p,n)$ to denote the distribution of $X$, while using $\pi(p|\alpha,\beta) = Beta(\alpha,\beta)$ to denote the distribution of $p$. May I ask why the notation $\pi$ is used here? Is there a specific meaning?

Answer 1

ISLR (An Introduction to Statistical Learning) states:

Let $π_k$ represent the overall or prior probability that a randomly chosen observation comes from the kth class.

This above is in reference to the the Bayes Theorem.

Bayes theorem is given by $Pr(Y = k|X = x) = \frac{π_kf_k(x)}{\sum_{l=1}^{K}π_lf_l(x)}$

Note that the meaning of $f_k(x)$ as the following:

Let $f_k(x) ≡ Pr(X = x|Y = k)$ denote the density function of X for an observation that comes from the kth class.

see page 138 and 139 (corresponds to the 152nd and 153rd page due to index etc.) in this link https://hastie.su.domains/ISLR2/ISLRv2_website.pdf