Transformation of a random variable (poisson distribution)

Question

Let $X \sim \text{Poisson} \left( \lambda \right)$

And $Y = \text{MIN} \left[X, r+1 \right]$

How can I construct pmf of $Y$?

I am looking for some generic approach so that I can apply such approach for any discrete or continuous random variable

Answer 1

You can construct the pmf from the CDF/ survival function for example.

Note that for a general discrete random variable $Z$ supported on the integers $$ P(Z=z) = P(Z>z-1) - P(Z>z) $$

In your case $P(Y > y) = P(X>y \quad \text{and}\quad r+1 >y)$ for $y=0,1,2,\dotsc r$. (why?)

and $$ P(Y=y) = P(Y>y-1) - P(Y>y) $$

for $y=0,1,2,\dotsc r+1$

where $P(X>x) =\sum_{k=x+1}^\infty\frac{\lambda ^k}{k!}$.