An accurate and soundly based representation of the complex stochastic processes encountered with optical preamplification in direct detection receivers is obtained. The objective of this study is the identification of a comprehensive moment generating function (MGF) characterisation for practical optically preamplified receivers. To effect this we build on a skeleton abstract formulation outlined some time ago with only limited physical and structural interpretation by Personick. Here we develop a MGF for the postdetection noise processes which takes into account a comprehensive set of key amplifier and receiver parameters, including gain and spontaneous noise characteristics of the semiconductor laser amplifier (SLA), amplifier facet reflectivity, bandwidth of the associated optical filter and postdetection electrical receiver filter characteristics. This comprehensive MGF may be used to obtain, via an independent route, the simple mean and variance characterisation of Yamamoto. We demonstrate this correspondence on a term-by-term basis and identify also minor discrepancies in two terms of the Yamamoto noise representation. The importance of the new MGF formulation is that it provides, for the first time, a sound formal basis which may be adopted for receiver design and analysis; this will enable, in any given situation, the adequacy or otherwise of a simple Gaussian approximation to be assessed.