In this paper, we present the end-to-end distance of randomly self-paired polymers (RSPPs). We define a randomly self-paired polymer as a linear polymer each of whose monomers has a probability, f (0<f<1), of pairing with any other one monomer. The RSPP model is inspired by numerous observations that the ends of RNAs are in close proximity. We use this model to explain this proximity. The prediction made by the RSPP model is consistent with these observations. Mapping an RNA with a length of 1000 nucleotides and a pairing fraction of 0.6 onto our RSPP model, for example, we predict an expected end-to-end distance of about 14 unpaired bases. We have also found that the expected end-to-end distance of the RSPP scales roughly as the 1/4 power of its total length.