This paper investigates the response of a bistable energy harvester to random excitations that can be approximated by a white noise process. Statistical linearization (SL), direct numerical integration of the stochastic differential equations, and finite element (FE) solution of the Fokker–Plank–Kolmogorov (FPK) equation are utilized to understand how the shape of the potential energy function influences the mean output power of the harvester. It is observed that, both of the FE solution and the direct numerical integration provide close predictions for the mean power regardless of the shape of the potential energy function. SL, on the other hand, yields nonunique and erroneous predictions unless the potential energy function has shallow potential wells. It is shown that the mean power exhibits a maximum value at an optimal potential shape. This optimal shape is not directly related to the shape that maximizes the mean square displacement even when the time constant ratio, i.e., ratio between the time constants of the mechanical and electrical systems is small. Maximizing the mean square displacement yields a potential shape with a global maximum (unstable potential) for any value of the time constant ratio and any noise intensity, whereas maximizing the average power yields a bistable potential which possesses deeper potential wells for larger noise intensities and vise versa. Away from the optimal shape, the average power drops significantly highlighting the importance of characterizing the noise intensity of the vibration source prior to designing a bistable harvester for the purpose of harnessing energy from white noise excitations. Furthermore, it is demonstrated that, the optimal time constant ratio is not necessarily small which challenges previous conceptions that a bistable harvester provides better output power when the time constant ratio is small. While maximum variation of the mean power with the nonlinearity occurs for smaller values of the time constant ratio, this does not necessarily correspond to the optimal performance of the harvester.