Verified Certificate

Supported by the following organizations

This is to certify that Aaron Arrese successfully completed and received a passing grade in 6.431x: Probability - The Science of Uncertainty and Data a course of study offered by MITx, an online learning initiative of the Massachusetts Institute of Technology.

Noted by

John Tsitsiklis

John Tsitsiklis

Professor, Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology

Krishna Rajagopal

Krishna Rajagopal

Dean for Digital Learning Massachusetts Institute of Technology

About edX

edX offers interactive online classes and MOOCs from the world’s best universities, including MIT, Harvard, Berkeley, University of Texas, and many others. edX is a mission driven initiative created by founding partners Harvard and MIT.

About edX Verified Certificates

An edX Verified Certificate signifies that the learner has agreed to abide by the edX honor code and completed all of the required tasks of this course under its guidelines, as well as having their photo ID checked to verify their identity.