1 The Poisson is discrete, in this case it's the number of calls per unit of time
2 The Exponential is continuous, it's the time between calls.
Then means that we had zero arrivals by time t.
Let Y be the time of the first arrival (this is the exponential rv).
means the same, we have had no arrivals by time y.
But that is the distribution of an exponential, with parameter
So in (a) you want no calls in any 30 minute period which is twice the mean