Beta distribution
A good interpretation of Beta distribution can be referred to a thread from What is the intuition behind beta distribution
The short version is that the Beta distribution can be understood as representing a distribution of probabilities, that is, it represents all the possible values of a probability when we don't know what that probability is.
For \(\theta\sim Beta(\alpha,\beta)\) \[ \begin{gather*} p(\theta)=\frac{\theta^{\alpha-1}(1-\theta)^{\beta-1}}{B(\alpha,\beta)}\\ =\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\theta^{\alpha-1}(1-\theta)^{\beta-1}\\ \theta\in[0,1] \end{gather*} \] The expectation and variance of Beta distribution is \[ \begin{gather*} E(\theta)=\frac{\alpha}{\alpha+\beta}\\ var(\theta)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} \end{gather*} \] When \(\alpha=\beta=1\), Beta distribution is equivalent to the uniform distribution \[ \begin{gather*} p(\theta)=\frac{\Gamma(2)}{\Gamma(1)\Gamma(1)}\theta^0(1-\theta)^0\\ =\frac{1\Gamma(1)}{1*1}\\ =1\\ F_{\Theta}(x)=\int_0^x 1d\theta=x\in[0,1]\\ \Rightarrow \theta\sim U(0,1) \end{gather*} \]