Let $X_1, X_2, …, X_n$ be a random sample of a random variable X, which has Gamma distribution with parameters $α = 3$ and $β$.
a) Find maximum likelihood estimator (MLE) of $β$.
b) The weekly downtime X (in hours) for a certain industrial machine has Gamma distribution with parameters $α = 3$ and $β$. The loss, in dollars, to the industrial operation as a result of this downtime is given by $L = 30X + 2X^2$. Find MLE of expected loss.
I found the answer for part a to be $β=\frac{\bar{x}}{3}$ and it seems right since the expected value for gamma distribution is $E[X] = αβ$.
But I am not sure how to deal with part b. Only thing I can think of is finding $E[L]=E[X^2+30X]$, taking the derivative of $ln(E[L])$ with respect to $β$ and setting it to $0$. This gives me an estimator for $β$, but as I understand the question, I need to find an estimator for expected loss. Can you help me?