For a class of large-scale separable convex optimization problems, an adaptive stochastic primal-dual algorithm is proposed. The optimization problem is reformulated as a saddle point problem with separable dual variables. Then, the dual variables of the saddle point problem are randomly updated with adaptively selected step size. The adaptive stochastic primal-dual algorithm almost surely converges with rate O〖JB((〗〖SX(〗1〖〗N〖SX)〗〖JB))〗 in an ergodic sense. The results of numerical experiments indicate that the algorithm can effectively solve the problem of positron emission computed tomography.