Find the MLE for Lamda and Theta

f(x|Lambda, Theta) = Lambda*exp(-Lambda(x-Theta))Indicator[x>=Theta]

I got Lambda-hat = (Theta + 1)/Xbar

and Theta-hat is nLambda=0, so lambda = 0 (Which seems like it must be wrong)

but assuming It was correct I just plugged in lambda = 0 into the MLE for lambda hat (not even sure if this is what you are supposed to do) and got Theta = -1.

I get the feeling im compltetly wrong can someone check this, I can't find where I have gone wrong.

Thanks