To calculate the characteristic function, mean, and variance from a given probability density function (PDF) p(x), standard formulas can be applied, such as the characteristic function φ(t) = ∫ e^(itx)f(x)dx and moments m_k = ∫ x^kf(x)dx. A uniform distribution example was discussed, where p(x) = 1/2b, leading to confusion about the correct form of the characteristic function. The conversation highlighted the importance of understanding the relationship between the characteristic function and its implications for calculating moments and variance. Ultimately, the discussion emphasized the need for clarity in notation and the application of Fourier transform properties in statistical calculations.