A Review of Activity Time Distributions in Risk Analysis <s> 12) Tilted Beta Distribution <s> THE usual procedure in obtaining theoretical distributions is to obtain a statement of the frequency function, f(x). The distribution function, F(x), may be expressed as a definite integral. One may adopt the opposite viewpoint and obtain a distribution function, F(x), from which the associated frequency function, f(x), may be obtained by differentiation. F(x) will be referred to as a cumulative frequency function (c.f.f.). The latter viewpoint was discussed by Burr in [1]. It is the purpose of this paper to give a family of c.f.f.'s which is believed to be new, which is readily handled from the calculational standpoint, which has useful values of a3 and a4, the standard third and fourth moments, including the range of values for failure data mentioned below, and which yields J-shaped frequency functions. A graph is presented which shows the values of a32 and <s> BIB001 </s> A Review of Activity Time Distributions in Risk Analysis <s> 12) Tilted Beta Distribution <s> Recent years have seen an increase in the development of robust approaches for stochastic project management methodologies such as PERT (Program Evaluation and Review Technique). These robust approaches allow for elevated likelihoods of outlying events, thereby widening interval estimates of project completion times. However, little attention has been paid to the fact that outlying events and/or expert judgments may be asymmetric. We propose the tilted beta distribution which permits both elevated likelihoods of outlying events as well as an asymmetric representation of these events. We examine the use of the tilted beta distribution in PERT with respect to other project management distributions. <s> BIB002
Hahn and Martín BIB002 introduced the tilted beta distribution with probability density function The mean and variance of the tilted beta distribution are ( ) respectively. The distribution is a mixture of the tilting distribution BIB001 r v − = . The mixing parameter θ was elicited as a judgmental estimate as in Hahn  . The tilted-beta distribution accommodates outlying events.