In this paper, we consider the function f p ( t )= 2p X 2 ( 2p t+p;p ) , where χ²(x; n) defined by X 2 ( x;p )= 2 −p/2 Γ( p/2 ) e −x/2 x p/2−1 , is the density function of a χ²-distribution with n ...

Probability theory forms the mathematical backbone for quantifying uncertainty and random events, providing a rigorous language with which to describe both everyday phenomena and complex scientific ...

Continuity or discontinuity of probability density functions of data often plays a fundamental role in empirical economic analysis. For example, for identification and inference of causal effects in ...