Expression for Probability of Being Between Two Poisson Random Variables? I have two independent Poisson random variables $A \sim \text{Poisson}(\lambda_A)$ and $B \sim \text{Poisson}(\lambda_B)$. For a fixed given integer $k$, I'd like to determine
$$P(A < k \leq A + B).$$
Is there an analytical expression for this probability as a function of $k, \lambda_A, \lambda_B$?

Edit 1: Is the following approach valid?
$$P(A < k \leq A + B) = 1 - P(A < k \leq A + B)^C $$
where $\cdot^C$ denotes the complement. The complement here is given by
$$P(A < k \leq A + B)^C = P(A \geq k) + P(k > A + B) $$
Plugging the complement in gives us:
\begin{align*}
P(A < k \leq A + B) &= 1 - P(A < k \leq A + B)^C\\
&= 1 - P(A \geq k) - P(k > A + B)\\
&= 1 - (1 - P(A < k)) - P(A + B < k)\\
&= P(A \leq k - 1) - P(A + B \leq k-1)
\end{align*}
where the final expression is the CDF of $A$ up to $k-1$ minus the CDF of $A+B$ up to $k-1$.
Is this correct? And if so, can I simplify further?
 A: For simplicity, you can write:
$$\begin{align}
\mathbb{P}(A < k \leqslant A+B)
&= \mathbb{P}(A+B \geqslant k) - \mathbb{P}(A \geqslant k) \\[6pt]
&= [1-\mathbb{P}(A+B \leqslant k-1)] - [1-\mathbb{P}(A \leqslant k-1)] \\[6pt]
&= \mathbb{P}(A \leqslant k-1) - \mathbb{P}(A+B \leqslant k-1) \\[6pt]
&= F_\text{Pois}(k-1 | \lambda_A) - F_\text{Pois}(k-1 | \lambda_A+\lambda_B). \\[6pt]
\end{align}$$
The CDF of the Poisson distribution does not simplify any more than writing it as a sum of the mass values, so this is really can't simplify any more than this.  (As whuber notes in the comments, you can "simplify" it to an integral if you prefer, but this will still require computer evaluation.)  You can compute this probability in R using standard probability functions for the Poisson distribution.  Here is an example taking $k=6$, $\lambda_A=3$ and $\lambda=6$.
#Set parameters
K <- 6
LAMBDA.A <- 3
LAMBDA.B <- 5

#Compute probability
ppois(K-1, lambda = LAMBDA.A + LAMBDA.B, lower.tail = FALSE) - 
  ppois(K-1, lambda = LAMBDA.A, lower.tail = FALSE)

[1] 0.724846

