The intrinsic electron anti-neutrino component in the muon anti-neutrino Tokai to Kamiokande (T2K) beam represents the largest irreducible background for the muon anti-neutrino to the electron anti-neutrino appearance measurement at Super-Kamiokande.

In this thesis, the electron anti-neutrino charged-current inclusive cross-section on carbon is measured using the T2K near detector (ND280). The measurement is performed in two steps: the selection of the electron anti-neutrino events, and the correction of the detector inefficiency and mis-reconstruction (unfolding).

In the selection step, a new particle identification based on the Boosted Decision Tree (BDT) was developed. The Particle IDentifications (PIDs) from the Fine-Grained Detectors (FGDs), Time Projection Chambers (TPCs) and Electromagnetic Calorimeters (ECALs) are used as input for the BDT to select the positron track; a characteristic feature of the charged-current interaction of the electron anti-neutrino.

The BDT parameters were tuned to maximize the selection efficiency, purity and Kolmogorov-Smirnov (KS) test. The uncertainty on the BDT output test statistic was calculated by propagating the input variables systematics using Monte Carlo (MC) simulation.

Moreover, we have developed an improved unfolding algorithm based on the D'Agostini 2010 method as well as a framework to test the unfolding performance.

In order to to constrain the gamma background, we have also constructed a control sample to select the gamma events and to use it in the simultaneous unfolding framework.

Finally, we measured the electron anti-neutrino charged-current inclusive cross-section on carbon with and without using the control sample.