A remark on the Δ function

Abstract

A probability distribution over an infinite interval is discussed. The probability density function is equal to zero everywhere, but its integral over the whole interval is equal to one. Some properties of the function are pointed out. The Δ function is treated as the limit case of a probability density function defined over an interval of length L. Unlike in a possible definition of the function δ of Dirac, the limit is taken with the length L approaching infinity. This way, the value of the Δ function tends to zero everywhere, but the integral of the function over [−∞, ∞] is equal to one. This is a generalized function that can be interpreted as a uniform density function defined on the whole independent variable axis.Some properties of the Δ function are discussed, like the cumulative function and superposition with other density functions. It is shown that the corresponding cumulative function is equal to 1/2 everywhere on (−∞, ∞). Treating Δ as a density function over (−∞, ∞), we can see that, while randomly shooting, the probability of hitting any point within a finite interval is equal to zero. Note, however, that such event is not impossible.