### Beta-Negative Binomial Experiment

#### Description

In this experiment, a random probability \( P \) has a beta distribution with left parameter \( a \) and right parameter \( b \). Then, Bernoulli trials with success probabiity \( P \) are performed. Random variable \( V \) is the trial number of the \( k \)th success, and has the beta-negative binomial distribution with parameters \( k \), \( a \), and \( b \). The top graph shows the sequence of Bernoulli trials in a timeline, with each success as a red dot and each failure as a green dot. The second distribution graph shows the probability density function and empirical probability density function of \( P \). The first distribution graph shows the probability density function and empirical probability density function of \( V \). The first table records the outome \( (P, V) \) of the experiment. The second table records the same information in text form as the graph above it. The parameters \( k \), \( a \), and \( b \) can be varied with input controls.