## Expected value of Poisson distribution with varying parameter

Suppose the number of babies born every hour follows a Poisson distribution with parameter K. If K varies hourly according to a uniform distribution on [100, 300], what is the expected number of births in a randomly chosen hour?

Since K follows a uniform distribution, can I just set K = 200 (the average) and say that E[# births] = 200, because the expected number of occurrences of a Poisson distribution is equal to the Poisson parameter? It doesn't feel right to me. I also feel like you might be able to set up some condition between the Poisson and the uniform distributions and perhaps use something like E[E[X|Y]] = E[X] to reduce it.