First of all, you have to pay attention to the fact that there is no equality in the relation you presented, but rather a proportionality. Meaning that the posterior is given by this relation up to a nonzero constant. Now, imagine that you have a random variable , the posterior at t is given by summing the values from to , where , multiplying the result by a constant then taking e to the power of the result. Obviously, some constraints should be imposed on the definition of X such that p becomes a proper probability function.
Assuming that you have defined X, T, and sigma properly in your code, you can use this to compute the posterior
p = arrayfun(@(t) exp(sum(X(t:T))*2./sigma^2), t);