The paper presents a detailed study on the thermal vibration of a single-walled carbon nanotube by using different beam models of continuum mechanics, together with the law of energy equipartition, and the molecular dynamics simulations. The basic finding of the study is the relation, derived by using the Timoshenko beam model and the law of energy equipartition, between the temperature and the root-of-mean-squared (RMS) amplitude of thermal vibration at any cross section of the carbon nanotube. The molecular dynamics simulations show that both the Euler beam model and the Timoshenko beam model can roughly predict the thermal vibration of lower order modes for a relatively long carbon nanotube. However, the Timoshenko beam model, compared with the Euler beam model, offers a much better prediction of the RMS amplitude of the thermal vibration near the fixed end of the carbon nanotube. For the thermal vibration of a relatively short carbon nanotube or higher order models of a relatively long carbon nanotube, the difference between the Timoshenko beam and the Euler beam in dynamic prediction becomes obvious, and the Timoshenko beam model works much better than the Euler beam model.