2 Maximum Likelihood [ 15 pts] Suppose we have n i.i.d (independent and identically distributed) data samples from the following probability distribution. This problem asks you to build a log-likelihood function, and find the maximum likelihood estimator of the parameter(s). (a) Exponential distribution [ 5 pts] The exponential distribution is defined as P(x)=1ex,with0x1 assume that x0>0 is given. Find the MLE of . (c) Normal linear regression model [5 pts] The regression equations can be written in matrix form as y=X+ where y is the N1 vector of observations of the dependent variable, X is the NK matrix of regressors, and is the N1 error terms. With the i.i.d assumption, multivariate normal distribution of on X, and full rankX, we can construct that the likelihood function of the linear regression model is L(,2;y,X)=(22)N/2exp(221i=1N(yixi)2) Show that the MLE of the regression coefficients and the variance of the error terms 2 are ^N^N2=(XTX)1XTy=N1i=1N(yixi^N)