Suppose that $X_{1}, X_{2}, \ldots, X_{n}$ denote a random sample from an exponentially distributed population with mean $\beta$. Find the MLE of the population variance $\beta^{2}$.
My approach: I know that
Method of Maximum Likelihood: Suppose that the likelihood depends on $k$ parameters $\theta_{1},\theta_{2},\ldots,\theta_{k}$. Choose those values of the parameters that maximize the likelihood $\mathbb{L}[y_{1},y_{2},\ldots,y_{n}|\theta_{1},\theta_{2}\ldots,\theta_{k}]$.
So, in my problem a need to find the value of the parameter $\beta^{2}$ sucht that maximize the $$\mathbb{L}[x_{1},x_{2},\ldots,x_{n}|\beta^{2}]:=\mathbb{L}[\beta^{2}]$$
But I know that if $t(\theta)$ is a one-to-one function of $\theta$ and if $\hat{\theta}$ is the MLE for $\theta$, so the MLE of $t(\theta)$ is given by $$\hat{t(\theta)}=t(\hat{\theta})$$ This result, sometimes refered to as the invariance property of MLEs.
Now, using that result, I think I can calculate first $$\mathbb{L}(y_{1},y_{2},\ldots,y_{n}|\beta)=f(y_{1}|\beta)\times f(y_{2}|\beta)\times \cdots f(y_{n}|\beta)$$ where $f(y_{i}|\beta)$ is the density functio of the $Y_{i}\sim \mathbf{Exp}(\beta)$.
So, we have \begin{eqnarray*} \mathbb{L}(\beta)&=&\left(\frac{1}{\beta}\right)^{n}\mathbf{exp}\left[\frac{-y_{1}-y_{2}-\cdots -y_{n}}{\beta}\right] \end{eqnarray*}
Question: How can I write the support of $\mathbb{L}[\beta]$?
Also, I know to do $\frac{d\mathbb{L}[\beta]}{d \beta}=0$, and so we have $$\frac{d}{d\beta}\left(\left(\frac{1}{\beta}\right)^{n}\mathbf{exp}\left[\frac{-y_{1}-y_{2}-\cdots -y_{n}}{\beta}\right] \right)=0$$
since that $\ln(\mathbb{L}[\beta]$ and $\mathbb{L}[\beta]$ reach the maximum in the same $\beta$, by monotony, then we can calculate $\beta$ in $$\frac{\ln(\mathbb{L}[\beta])}{d\beta}=0 \implies \frac{1}{\beta^{2}}\sum_{1\leq i \leq n}y_{i}=\frac{n}{\beta} \implies \beta=\frac{1}{n}\sum_{1\leq i \leq n}y_{i}=\overline{Y}_n$$ So, we have that $$\hat{\beta}_{MLE}=\overline{Y}_n$$Now, by invariance property we can see that $$\hat{\beta}^{2}_{MLE}=\overline{Y}_n^{2}$$
Is correct my solution? Any suggestion?