At a 911 call center, calls come in at an average rate of one call every two minutes. Assume that the time that elapses from one call to the next has the exponential distribution.

Question

On average, how much time occurs between five consecutive calls?

Ninety-percent of all calls occur within how many minutes of the previous call?

Suppose that two minutes have elapsed since the last call. Find the probability that the next call will occur within the next minute. Answer

Answer 1

First, find the exponential distribution. Since the average call comes in 2 minutes. That's the average. This means that λ=1/2. So the exponential distribution is p(x) = .5e^-.5x and the CDF = 1 - e^-.5x

A) Because E(X) = 2 minutes, E(5X) = 5E(X) = 5•2 = 10 minutes.

B) Here we need to solve for x where CDF = 0.90.

1 - e^-.5x = 0.9

e^-.5x = 0.1

.5x = ln(10)

x = 2ln(10) ≈ 4.6052

This means that 90% of calls come within 4 minutes and 36.31 seconds.

C) Because each call is independent, the time to the next call is not affected by the previous time. This means we simply plug in x=1 into the CDF to find the probability of the call coming in within 1 minute.

P = 1-e^-.5 ≈ 0.3935

So there's a 39.35% chance the next phone call comes within a minute.