# Need some interpretation with plain English for a part in Bayesian Statistics with Beta proability distribution? [duplicate]

Can somebody explain why equation (6.3) and (6.4) are shown in the book and what the author is trying to say?

It feels to me that I am reading the text but I don't think I getting the true meaning that the author is trying to say.

I am reading the book, "Doing Bayesian Data Analysis" and the page is "127". • The beta distribution is commonly used in bayesian analysis as a prior distribution when the parameter of the model is a probability. See stats.stackexchange.com/search?q=beta+prior Dec 18, 2019 at 16:45

• A beta distribution has density proportional to $$x^{a-1}(1-x)^{b-1}$$ where $$a$$ and $$b$$ are parameters. Setting $$a=b=1$$ yields a uniform, since the density is constant.
• Like all PDF's, these ones must have a total probability of 1. To get this to work, you have to divide $$x^{a-1}(1-x)^{b-1}$$ by its own integral between 0 and 1. This integral, seen as a function of $$a, b$$, has a name because it has other uses in mathematics. It is called the Beta Function.
• Here's a fun fact to help develop intuitions about the beta distribution. If you simulate $n$ iid random variables from a uniform distribution on [0,1], then retain the $k$th smallest one, it has a Beta distribution with parameters $k$ and $n+1-k$. If $n$ is 5 and $k$ is 4, the density is proportional to $x^4(1-x)^2$, so it's bigger at 0.999 than it is at 0.001, and generally biased toward big values. This makes sense because you took one of the bigger numbers from your list. This type of theory can be used to study properties of the sample median, which is a commonly-used robust estimator. Dec 18, 2019 at 16:52
• I have read your sentences repeatedly but I get stuck even with the first thing, "If you simulate n iid random variables from a uniform distribution on [0, 1], then retain the $k_{th}$ smallest one, it has a Beta distribution with parameters k and $n+1-k$. I don't know what you mean by "simulate n iid random variables", it is a "beta distribution", and what do you mean by $k$ parameters and $n+1-k$. Did you mean "observations(data)" by "variables"?And why it is bigger than "0.999"? Apologies for my ignorance. Dec 18, 2019 at 17:56