# If $X\sim U(0,n)$, how can I show that $X-[X]\sim U(0,1)$?

If $$X\sim U(0,n) ; n \in \mathbb N$$ , how can I show that the distribution of $$Y =X-[X]$$ is $$U(0,1)$$?

Any hint will also help me...

• You should show your work/share your thoughts in your questions. – StubbornAtom Dec 20 '18 at 14:46

For $$Y=X-[X]$$, and $$0 $$P(Y Hence $$Y\sim U(0,1).$$

• It probably would be helpful to use align so that the key step is better readable. – Just_to_Answer Dec 29 '18 at 2:37

Hint :

If for all $$\phi$$ continuous bounded functions :

$$E[\phi(Y)]=\int_{\mathbb{R}} \phi(y) g(y) dy.$$

Then $$g$$ is the pdf of $$Y$$.

First step :

$$E[\phi(Y)]=\int_{\mathbb{R}} \phi(x-[x]) f_X(x) dx.$$

With $$f_X$$ the pdf of $$X$$.

Can you finish ?