Can someone please show me how to do this question
Given an observation $\displaystyle u>0$ of $\displaystyle Y$ the likelihood of $\displaystyle \theta$ is:
$\displaystyle l(\theta|u)=2u\theta e^{-\theta u^2}$
and this is a maximum when:
$\displaystyle \frac{\partial}{\partial \theta}l(\theta|u)=0$
which you solve for $\displaystyle \theta$ in terms of $\displaystyle u$ to get the maximum likelihood estimator.
If you have a sample of size bigger than one, the likelihood is the product of the individual sample value likelihoods, and it will be easier to maximise the log likelihood, but that will give the same result as maximising the likelihood.
CB