Finding a PMF with random variables

Question

Let $X$ be a discrete random variable that is uniformly distributed over the set of integers in the range $[a,b],$ where $a$ and $b$ are integers with $a<0<b$. Find the PMF of the random variables $\max\{0,X\}$ and $\min\{0,X\}$

Answer 1

There are $b-a+1$ integers between $a$ and $b$, inclusive. The number between $a$ and $0$, inclusive, is $1-a$. The remaining $b$ numbers are $\gt 0$.

Let $Y=\max(0,X)$. Then $Y=0$ precisely if $X\le 0$. Thus

$$\Pr(Y=0)=\frac{1-a}{b-a+1}.$$

If $X$ is positive, then $Y=X$. So for any $y$ with $1\le y\le b$ we have $\Pr(Y=y)=\frac{1}{b-a+1}$.

Now it's your turn for the distribution of $Z=\min(0<X)$.