This chapter first considers the existing academic evidence that private equity-backed companies outperform their peers and then looks at standard explanations for this outperformance.It is argued that existing explanations are inadequate.

In the previous chapter, we introduced distributions and density functions as two alternative methods to characterize the randomness of variates. In this chapter, we introduce the final method considered in this book, and relate it to the previous two: the moments of a variate (including mean, variance, and higher-order moments). The condition for the existence of moments is explained and justified mathematically. These moments can be summarized by means of "generating functions". We define moment-generating functions (m.g.f.s). Cumulants and their generating functions are introduced. Characteristic functions (c.f.s) always exist, even if some moments do not, and they identify uniquely the distribution of the variate, so we define c.f.s and the inversion theorem required to transform them into the c.d.f. of a variate. We also study the main inequalities satisfied by moments, such as those resulting from transformations (Jensen) or from comparing moments to probabilities (Markov, Chebyshev). We also show that the mean, median, and mode need not be linked by inequalities, as previously thought.