The aim of this thesis is to study con dence regions under the Inverse Gaussian Distribution and how it applies to the construction of con dence ellipses of parameters associated with the Inverse Gaussian Distribution. This thesis consists of two parts, the rst part being the theoretical with where we seek to operate a classical estimation method known as the maximum likelihood. This is achieved by deriving maximum likelihood equations and estimating the Fisher Information matrix for the construction of a normal approximation which gives ellip- tical con dence regions by approximating the Inverse Gaussian Distribution. The second part which is the computational centres on constructing asymptotic con dence ellipses for parameters of the Inverse Gaussian Distribution and using the coverage probabilities to compare with the con dence coe cient of 0.98. We then apply the Monte Carlo method using four cases of sample size n 30, 100, 500 and 1,000 and six values of and the parameter is particularized and set to be 1. The statistical software R is then used for our simulation technique with 10,000 iterations. The study concluded that the coverage probabilities of con dence ellipses for pa- rameters of the Inverse Gaussian Distribution increase when sample sizes n increase and they are close to the con dence coe cient of 0.98 for all values of both param- eters. Additionally, all chosen values of parameters and of the Inverse Gaussian Distribution give high coverage probabilities when n is large.