# Understanding parameters of Beta distribution [duplicate]

I've heard that the $\alpha$ and $\beta$ parameters of the Beta distribution intuitively represent the number of successes and failures, respectively.

1) If so, what's the purpose of subtracting $1$ from them in the exponent?

2) If so, why would then $\alpha=1$ and $\beta=1$ represent a uniform distribution? Do we assume that there is one success and failure initially? Does this have anything to do with Laplace's Rule of Succession?

3) How can we intuitively understand positive real-valued shape parameters for the distribution?

For 1), it is not a valid density integrating to 1 otherwise. Of course, one could let $\tilde\alpha=\alpha-1$ and $\tilde\beta=\beta-1$ and write $$\pi \left( \theta \right) =\frac{\Gamma \left( \tilde\alpha+\tilde\beta+2\right)}{\Gamma \left( \tilde\alpha+1\right) \Gamma \left( \tilde\beta+1\right) }\theta^{\tilde\alpha}\left( 1-\theta \right) ^{\tilde\beta}$$
For 2), just plug in: the beta density $$\pi \left( \theta \right) =\frac{\Gamma \left( \alpha+\beta\right)}{\Gamma \left( \alpha\right) \Gamma \left( \beta\right) }\theta^{\alpha-1}\left( 1-\theta \right) ^{\beta-1}$$ becomes $$\pi \left( \theta \right) =\frac{\Gamma \left( 2\right)}{\Gamma \left( 1\right) \Gamma \left(1\right) }\theta^{1-1}\left( 1-\theta \right) ^{1-1} =1$$
• 1) Yes, but it's already restricted to an interval and normalized. Could we not just remove the $-1$ and renormalize it? Is it simply a matter of mathematical convenience? i.e. use of the $\Gamma$ function 2) I know mathematically it works out, but is there any way we can interpret using $(1,1)$ to represent a uniform distribution in the context of successes and failures? 3) What would a non-integer $n$ mean in the "fictitious sample interpretation"? Edit: 2) After looking at your answer again, you mention that it represents $\alpha_{0}-1$ successes, so $0$ successes.