Stats/Probability - Poisson Distribution - How do I calculate the probability between two points in R?

Question

I need to find P(400<=X<=500) in one year using R

Given:

Poisson Distribution

Lambda = E(X) = 1.3 crimes per day = 474.5 per year

The only methods I can find online are to manually input numbers 400-500 using dpois, I really have no idea how to use R very well so if someone could help me that would be much appreciated.

relevant question outlined in red

Answer 1

Consider the following three events:

1. $A: X \lt 400$
2. $B: X \leq 500$
3. $C: 400 \leq X \leq 500$

Note that $A$ and $C$ are mutually exclusive, and that $A \cup C = B$. So if you can find the probability of $A$ and $C$, you should be able to find the probability of $B$.

To find those probabilities in R, notice that when you look up distributions in the help file that there's usually one function starting with d that gives the density function, and another starting with p that gives the cumulative density, which may be more useful here.