Find an analytical expression for the cumulative distribution function of the random variable 1/(1+((1/p-1)(1-c))/s) given that p, c, and s are independent random variables with beta distributions with known integer parameters?
This is the positive predictive distribution of a patient’s having a disease given its prevalence in the population is described by a beta distribution p and the patient has tested positive for the disease in a test having sensitivity and specificity described by beta distributions s and c respectively. We also need to compute the inverse function, i.e., the quantile function. We would like to be able to compute these functions with arbitrary accuracy, or with accuracy equivalent to that with which we can now compute beta distributions.
This would perhaps be an easy problem for APPL under Maple because the expression involves only additive shifts, negation, reciprocations, and multiplicative convolutions on beta distributions.