cdf of sum of a discrete and continuous random variable Let U be uniformly distributed on the interval $(0, 2)$ and let V be an independent
random variable which has a discrete uniform distribution on $\{0, 1, . . . , n\}$. i.e.
$P\{V = i\} =\frac{1}{n+1}$ for $i = 0, 1, . . . , n.$
Find the cumulative distribution function of $X = U + V$ .
 A: Hint: In order to see what is going on, it is useful to give $n$ a concrete value, like $20$. Informally, we have a $21$-sided fair die with the numbers $0$ to $20$ written on its faces. We roll the die, and add to the result $V$ a number $U$, chosen from the interval $(0,2)$ with uniform distribution. 
The result is our random variable $X$. We want to find the cumulative distribution function $F_X(x)$ of $X$. So we want to find, for any $x$, the probability that $X\le x$. Let us work with a particular value of $x$, like $x=8.74$.  
Under what conditions is $X\le 8.74$? If $V$ takes on any of the values $0,1,2,3,4,5,6$, then for sure $X\le 8.74$. This is because with probability $1$, we have  $U\le 2$.
The probability that $V\le 6$ is $\frac{7}{21}$.
But $X$ is also $\le 8.74$ if $V=7$, and $U\le 1.74$. The probability that this happens is $\frac{1}{21}\cdot\frac{1.74}{2}$.
Finally, $X$ is also $\le 8.74$ if $V=8$, and $U\le 0.74$. The probability that this happens is $\frac{1}{21}\cdot\frac{0.74}{2}$.
For the probability that $X\le 8.74$, add up. We get
$$\Pr(X\le 8.74)=\frac{7}{21}+\frac{1}{21}\cdot\frac{1.74}{2}+\frac{1}{21}\cdot\frac{0.74}{2}.$$
The same reasoning will give you the cumulative distribution function $F_X(x)$ for any $x$ between $0$ and $n+2$. For completeness, one should add that $F_X(x)=0$ if $x\lt 0$, and $F_X(x)=1$ if $x\ge n+2$. 
Let's start doing the general case. What did we do with $x=8.74$? First we took the integer part $\lfloor x\rfloor=\lfloor8.74\rfloor$. Then we backed up by $2$ to $\lfloor x\rfloor -2$. If $V\le \lfloor x\rfloor -2$, then for sure $X\le x$. But $X$ can also be $\le x$ in a couple of other ways: (i) if $V=\lfloor x\rfloor -1$ and $U$ is of suitable size or (ii) if $V=\lfloor x\rfloor$ and $U$ is of suitable size.  
