So the problem asks:
A zoo owns a population of rare Galapagos turtles. The turtles bring in a lot of visitors but unfortunately do not breed in captivity. To maintain the population, new turtles are brought in at a rate of 0.3/year. The scientists are still not sure about the mortality of these species, so for simplicity the zookeeper assumes that the longevity of each turtle has exponential distribution with mean $30$ years. Find the long-run proportion of time when there are two or fewer turtles in the zoo.
My approach:
from the given information, the longevity of each turtle has exponential distribution with mean = 1/$\theta$ = $30$ yrs
Therefore, the longevity follows exponential distribution with parameter $\theta$ $= 1/30=0.033yrs $
By the probability distribution of exponential distribution, $p(x≤2)=p(x=0)+p(x=1)+p(x=2)$
$=0.033e^{-0.033*0}+0.033e^{-0.033*1}+0.033e^{-0.033*2}$
$=0.0639$
Is this the right approach to do these kinds of problems?