Exploring Beta-Like Distributions

The most well known probability distribution of probabilities is the Beta distribution. If we have observed $r$ `successes', each having a probability $\theta$, and $n-r$ `failures', each having a probability $1-\theta$. In this paper we will derive a whole family of Beta-like distributions, which take as their data not only the number of successes and failures, but also values on predictor variables and time to failure or time without failure.