Distribution function of uniform distribution

Question

I don't know why the distribution of this question is that when x is in between 0 and theta. In solution.. is that right? I searched the distribution of uniform distribution. But it is alike with that but contains gauss symbol. I think the difference is caused by 'maximum'. anyway i want to know only when x is in between 0 and theta in solution

Answer 1

The maximum is less than $x$ iff each variable is less than $x$. For $0\le x< \theta$ $$P[X_{(n)}\le x]=P[X_1\le x,X_2\le x,...,X_n\le x]=\prod_{j=1}^nP[X_j\le x]\\=\prod_{j=1}^n\int_0^xf_{X_j}(t)dt=\prod_{j=1}^n\int_0^x{1\over\theta}dt=\prod_{j=1}^n\left({x\over\theta}\right)=\left({x\over\theta}\right)^n$$