Interesting probability distribution of a mixed type random variable $Y$ Let $X$ and $U$ be independent random variables with:
$$P(X=k)=\frac1{N+1} \text{ for } k=0,1,2,\ldots,N$$
and $U$ having uniform $(0,1)$ distribution.
Let $Y=X+U$.
Find distribution function of $Y$.
I have tried to solve the problem by conditioning on value of $X$ and making use of total probability theorem.
I have got $P(Y\le y)=y-\frac N2$.  Is it correct? Please help.

It is a uniform distribution on $[0, N+1]$. Just take $k\leq a<b\leq k+1$ and calculate $P(Y\in [a,b))$.
