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We derive expressions for the effect of mechanical thermal noise on a vibrational microelectromechanical system gyroscope, including the angle of random walk, the noise equivalent rotation rate, and the spectral density of the noise component of the rate measurement. We explicitly calculate and compare the output signal due to rotation and the output due to noise. We avoid several ambiguities in the literature concerning bandwidth and correctly observe a factor of two reduction in noise power due to synchronous demodulation. We use stochastic averaging to obtain an approximate "slow" system that clarifies the effect of thermal noise and shows the effect of frequency mismatch between the drive and sense axes. We compute the noise equivalent rate for both open-loop and force-to-rebalance operation of the gyroscope.