A bird can fly and walk on the ground. It has to reach a certain destination. When it flies it has a certain PDF given as: $$P_{fly}(x,y,t)$$ where 2-D motion (x,y) is considered and $t$ is time. On the other hand, when it walks it has the PDF as $$P_{walk}(x,0,t)$$ I have to find the first arrival time distribution of this bird. For a single PDF (either walking or flying) I can find the PDF of arrival time. However, I do not how to proceed for such a compound problem?

The approach that I take:

I model the motion of the bird as a two-state Markov chain, as it can be in a state of walking or flying. I assume that if the bird walks it remains in state walking for that time and if it flies then it remains in state flying. Suppose, I also have the transition probabilities. But I do not know how to proceed further. Also, if I am considering walking/flying as a state then its parameters (x,y) always change.


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