Relation between P.D.F and Wavelets

Abstract

This paper proposes a relation between continuous probability density function and wavelets. Here we proved that, derivatives of probability density functions of continuous distribution are continuous wavelets satisfying conditions of wavelets. This proof is given completely in mathematical formation, and here we took basic concepts of probability distribution and Holders inequality to prove this theorem. And we took two examples to show the proof, 1)p.d.f of t-distribution for degrees of freedom 2 and 2).p.d.f of logistic distribution. These derivatives form new continuous wavelet family. Here we took p.d.f function and those successive derivatives, which are again continuous by fundamental theorem of calculus will become continuous wavelets by satisfying the conditions of the wavelets without FIR filters and without scaling function like Mexican and Morlet. Wavelet analysis has attracted attention for its ability to analyze rapidly changing transient signals.