Expected value of the number of experiments required to obtain the first success?

Question

Two independent observations $X_1$ and $X_2$ are generated from the Poisson distribution with mean 1. The experiment is said to be successful if $X_1 + X_2$ is odd. What is the expected value of the number of experiments required to obtain the first success? Help to solve this problem.

Answer 1

You can actually solve it with an MC approach. The starting state, $\emptyset$ and the abosrbent state is $S_{odd/even, even/odd}$. This is because the only way to get an odd sum is for $X_1$ to be even and $X_2$ odd or vice versa. The probability of this event is $$ p_{\emptyset, 1} = P(X_1 = 2k+1)\times P(X_2=2k)\times 2 =2 e^{-2} \sum_{k \in \mathbb{Z} }\frac{1}{(2k+1)!} \times \sum_{k\in \mathbb{Z}}\frac{1}{(2k!)} $$ Now, to get the mean time until the state you need to compute $$ m_{\emptyset ,1 } = 1+ p_{\emptyset, 1} \times 0 + (1-p_{\emptyset, 1})\times m_{\emptyset ,1 } $$

Can you handle from here?