We present the generalized signal detection theory (GSDT), where familiarity is described by a sparse binomial distribution of binary node activity rather than by normal distribution of familiarity. Items are presented in a distributed representation, where each node receives either noise only, or signal and noise. An old response (i.e., a "yes" response) is made if at least one node receives signal plus noise that is larger than the activation threshold, and item variability is determined by the distribution of activated nodes as the threshold is varied. A distinct representation leads to better performance and a lower ratio of new to old item variability, than a more distributed and less distinct representations. Here we apply the GSDT to empirical data on verbal and olfactory memory and suggest that verbal memory relies on a distinct neural item representation, whereas olfactory memory has a fuzzy neural representation leading to poorer memory and inducing a larger ratio of new to old item variability.