So, I'm having problems learning for a data analysis lecture. The topic is the construction of the central interval as our lecturer calls it. The concept shall path the way to understand confidence & credible intervals. But I don't even get his concept and therefore wanted to ask for help. I will give an excerpt of the passage I don't get (it is in the context of a bionmial distribution where r is the possible outcome of a bernoulli experiment):To this end, we define the Central Probability Interval and SmallestProbability Interval containing a probability at least 1-$\alpha$.This  interval  is  constructed  by  defining  a  smallest  member  of  the  set  of possible outcomes, $r_{1}$, and a largest member, $r_{2}$, subject to the criterion that$$P(r<r_{1}) <= \alpha/2$$and $$P(r>r_{2}) <= \alpha/2$$ while maximizing these probablities.First of all, $\alpha$ seems to be an arbitrary factor because it isn't mentioned anytime before. After choosing such an $\alpha$, we look for two numbers $r_{1,2}$. These numbers are limits which, when r surpasses these number, will yield a probability smaller than $\alpha$/2. But what about the maximizing of the probability? What probability?Stating this precisely, we write:$$r_{1} = 0 \quad\text{if}\quad P(r=0|N,p) > \alpha/2\quad\text{ else}\quad r_{1} =  \sup_{r\in0,...,N}[\sum_{i=0}^{r}P(i|N,p) <= \alpha/2]+1$$and$$r_{2} = N\quad\text{if}\quad P(r=N|N,p) > \alpha/2\quad\text{ else}\quad r_{2} =  \sup_{r\in0,...,N}[\sum_{i=r}^{N}P(i|N,p) <= \alpha/2]-1$$The set of our results in our Central Interval is$$O^{C}_{1-\alpha}=\{ r_{1}, r_{1}+1,...r_{2}\}.$$Now, I cannot make anything of the equations simply because I don't know how to read and process them properly. Maybe someone has an idea or can give me a good example?Your help is much appreciated.Edit:I found a hint online and I think, now I get it.We construct an interval which contains a prbability 1-$\alpha$. Therefore, the probability of all r values outside the interval [$r_{1}$, $r_{2}$] has to be $\alpha$. If you split this symmetric you get the criterion stated above.Now as for the equations. They add up all the probabilities which contribute to an outcome smaller than $\alpha$/2. Taking the $\sup_{r \in 0,...,N}$ now gives the r value for which this sum is maximized. Then you have to add the +1 in order to take the next r which then is $r_{1}$.
Central Probability Interval
statistics;data