exponential distribution probability(exponential time inteval)

Question

Can someone guide me how to solve this question,i want to familiar with the exponential distribution theory.

for part (a) I think "there are no calls within a 30 min. interval" = "getting a call after 30 min"

so if X = time between calls.

for part(a) answer is value of P(X<30)

Is my thinking correct?

Answer 1

for part (a) I think "there are no calls within a 30 min. interval" = "getting a call after 30 min"

Not quite. No calls within a 30 minute interval includes cases where you get a call after 31 min, 32 min, etc.

Let's say at $t=0$, you receive a call and then $X = $ the time until you receive the second call. It should be clear that this is just a restatement of the problem, one that I hope will be easier for you to wrap your head around.

So for part a), you want $P(X>30)$.

Can you use my restatement to answer the remaining parts?