Assume a pure random sample ${X_1},{X_2},...,{X_n}$ is given from a uniform distribution on the interval $\left[ {0,\theta } \right]$ with $\theta $ the unknown model parameter. Hence, the density function ${f_x}$ of $X$ is given by ${f_x} = {1 \over \theta }$, with $0 < x < \theta $. Consider ${\rm{maximu}}{{\rm{m}}_{i = 1,...,n}}{X_i} = {X_{\left( n \right)}}$.
Show that the density of $Y = {{{X_{\left( n \right)}}} \over \theta }$ is given by ${f_y} = n{u^{n - 1}}$, with $0 < u < 1$
Please bear in mind I don't have a strong math background. My attempt went like this:
If ${f_x} = {1 \over \theta }$, then the cdf is given by $${F_x} = \int\limits_0^x {{1 \over \theta }dt} = {x \over \theta }$$ Since it's an iid sample, the joint cdf of maximum $X$ would be something like $${F_{{x_{\left( n \right)}}}} = {\left( {{x \over \theta }} \right)^n}$$ Then $$Y = {{{X_{\left( n \right)}}} \over \theta } = {{{{\left( {{x \over \theta }} \right)}^n}} \over \theta } = {{{x^n}} \over {{\theta ^{n + 1}}}}$$ Which means $${f_y} = {d \over {dx}}Y = {1 \over {{\theta ^{n + 1}}}}n{x^{n - 1}}$$
From where I can't progress. Not sure about my steps until then either.