Memristors allow computing in memory, which may be leveraged by deep neural network (DNN) accelerators to reduce energy footprint. However, such gains in energy efficiency come at the cost of noise on the computation results due to the analog nature of memristors. In this work, we introduce a theoretical framework to estimate the mean squared error (MSE) of a memristor-based DNN. We propose an efficient software implementation of this framework which is shown to be orders of magnitude faster than using Monte-Carlo simulations. Additionally, we study two different techniques for mapping convolutional layers to memristors and compare their relative impact on the mean squared error and its computation time. The accuracy of the proposed analysis is first evaluated on a simple regression problem, and then on a more complex classification task with a network capable of achieving high accuracy on the CIFAR-10 dataset, which shows that our method is efficient over practical up-to-date DNNs. The proposed framework is then used to perform a meta-heuristic optimization of the memristor maximal conductance value so as to minimize the energy usage