We present an efficient algorithm to compute tight upper bounds of collision\nprobability between two objects with positional uncertainties, whose error\ndistributions are represented with non-Gaussian forms. Our approach can handle\nnoisy datasets from depth sensors, whose distributions may correspond to\nTruncated Gaussian, Weighted Samples, or Truncated Gaussian Mixture Model. We\nderive tight probability bounds for convex shapes and extend them to non-convex\nshapes using hierarchical representations. We highlight the benefits of our\napproach over prior probabilistic collision detection algorithms in terms of\ntighter bounds ($10$x) and improved running time ($3$x). Moreover, we use our\ntight bounds to design an efficient and accurate motion planning algorithm for\na 7-DOF robot arm operating in tight scenarios with sensor and motion\nuncertainties.\n