suppose X has a standard Cauchy distribution. Then . This is not differentiable at t = 0, showing that the Cauchy distribution has no expectation. Also, the characteristic function of the sample mean of n independent observations has characteristic function , using the result from the previous section. This is the characteristic function of the standard Cauchy distribution: thus, the sample mean has the same distribution as the population itself. As a further example, suppose X follows a Gaussian distribution i.e. . Then and A similar calculation shows and is easier to carry out than applying the definition of expectation and using integration by parts to evaluate . The logarithm of a characteristic function is a cumulant generating function, which is useful for finding cumulants; some instead define the cumulant generating function as the logarithm of the moment-generating function, and call the logarithm of the characteristic function the second cumulant generating function. Data analysis Characteristic functions can be used as part of procedures for fitting probability distributions to samples of data. Cases where this provides a practicable option compared to other possibilities include fitting the stable distribution since closed form expressions for the density are not available which makes implementation of maximum likelihood estimation difficult. Estimation procedures are available which match the theoretical characteristic function to the empirical characteristic function, calculated from the data. Paulson et al. (1975) and Heathcote (1977) provide some theoretical background for such an estimation procedure. In addition, Yu (2004) describes applications of empirical characteristic functions to fit time series models where likelihood procedures are impractical. Empirical characteristic functions have also been used by Ansari et al. (2020) and Li et al. (2020) for training generative adversarial networks. Example The gamma distribution with scale parameter θ and a shape parameter k has the characteristic function Now suppose that we have with X and Y independent from each other, and we wish to know what the distribution of X + Y is. The characteristic functions are which by independence and the basic properties of characteristic function leads to This is the characteristic function of the gamma distribution scale parameter θ and shape parameter k1 + k2, and we therefore conclude The result can be expanded to n independent gamma distributed random variables with the same scale parameter and we get Entire characteristic functions As defined above, the argument of the characteristic function is treated as a real