Weighted multimodal family of distributions with sine and cosine weight functions

In this paper, the moment of various types of sine and cosine functions are derived for any random variable. For an arbitrary even probability density function, the sine and cosine moments are used to define new families of univariate multimodal probability density and their corresponding characteristic functions. For illustration, two weighted multimodal generalizations of the t distribution are investigated. Furthermore, a method of calculating some interesting improper integrals is also presented. Finally, an explicit expression of the probability density function of the sum of independent t-distributed random variables with odd degrees of freedom is derived.