What is the probability that two independent Poisson random variables with parameter $\lambda<\infty$ have the same value?
My solution:
\begin{align} P = \sum_{i=0}^{\infty} \left(\frac{e^{-\lambda T} (\lambda T)^i}{i !} \right)^2 = e^{-2\lambda T} \sum_{i=0}^{\infty} \frac{ (\lambda T)^{2i}}{(i !)^2} \end{align}
I don't know how to continue from here. How can I evaluate the above summation? Can you help?