Let $Y_1,...Y_n$ be a random sample from the density function given by $$f(y|\theta)=\begin{cases} \frac{y^3}{6\theta^4}e^{-y/\theta}, & y > >0 \\ 0, & \text{elsewhere.} \end{cases}$$ Find the MLE $\hat{\theta}$ of $\theta$.
I'm stuck on finding the likelihood function. I tried:
$$L(Y_1,...Y_n|\theta)=\frac{\prod^n_{i=1}y_i^3}{(6\theta^4)^n}e^{-\sum_{i=1}^ny_i/\theta}$$
Is this correct? If so, then is the following also correct?$$ln(L(Y_1,...,Y_n|\theta)=ln(\prod^n_{i=1}y_i^3)-ln(6^n\theta^{4n})-\frac{\sum_{i=1}^ny_i}{\theta}$$