Exploring new probability density functions related to Brownian local time

This work introduces a new family of probability density functions constructed from E [ L B ( 1 , x ) ] , the expectation of the Brownian local time L B ( 1 , x ) at time one. The proposed framework builds probabilistic models directly from local time - a pathwise quantity rarely exploited in density construction - thereby offering an original perspective on stochastic analysis. Links are established with the standard normal distribution N ( 0 , 1 ) and several gamma distributions. Cumulative distribution functions and characteristic functions are derived and their moments compared to those of N ( 0 , 1 ) . The occupation density formula provides the central theoretical foundation, connecting the sample-path behavior of Brownian motion to its probabilistic structure. Motivation arises from the observation that the law of any integrable random variable Y is completely characterized by the function G ( x ) = E [ ( Y − x ) + ] . The resulting methodology enriches the interface between stochastic process theory and classical probability, with potential applications in quantitative finance, statistical physics, and the modeling of time-dependent random phenomena.