Conditional Beta Approximation: Two Applications - Habibi, Reza (Indonesian Journal of Mathematics and Applications, 2024)
Suppose that X,Y are two independent positive continuous random variables. Let P=X/(X+Y) and Z=X+Y. If X, Y have gamma distributions with the same scale parameter, then P distribution will be beta and P, Z are independent.  In the case that the distributions of these two variables are not gamma, the P distribution is well approximated by the beta distribution. However, P,Z are dependent. According to matching moment method, it is necessary to compute the moments of conditional distribution for beta fitting. In this paper, some new methods for computing moments of conditional distribution of P given Z are proposed. First of all, it is suggested to consider the regression method. Then Monte Carlo simulation is advised. The Bayesian posterior distribution of P is suggested. Applications of differential equations are also reviewed. These results are applied in two applications namely variance change point detection and winning percentage of gambling game are proposed. The probability of change in variance in a sequence of variables, as a leading indicator of possible change, is proposed. Similarly, the probability of winning in a sequential gambling framework is proposed. The optimal time to exit of gambling game is proposed. A game theoretic approach to problem of optimal exit time is proposed. In all cases, beta approximations are proposed. Finally, a conclusion section is also given.