Simple proofs of two results on convolutions of unimodal distributions

We give simple proofs of two results about convolutions of unimodal distributions. The first of these results states that the convolution of two symmetric unimodal distributions on is unimodal. The other result states that symmetrization of a unimodal random variable gives a symmetric unimodal random variable. Both our proofs avoid Khintchine's representation of a random variable that is unimodal about zero, and use the integral representation of the expectation of a non-negative random variable with its tail probability as the integrand.Convolution Khintchine's representation Symmetry Unimodality