The maximum likelihood estimator (MLE) under the parametric set-up with the right-censored (RC) data rarely has a closed form solution. The exponential distribution is an exception. Moreover, the parametric MLE’s with RC regression data are currently computed by iterative algorithms. The author shows that the MLE of the exponential distribution under the linear regression model with RC data has a closed form solution and prove its consistency. The author also proposes a diagnostic plot method and a test of H0: Y = bX +W and X and W are independent, where W has an exponential distribution, and apply them to a cancer data set.