1. ## Exponential distribution qn

Hi, could someone help me out with this question?

Calls to an emergency ambulance service in a large city are modeled by a Poisson process with an average 3.2 calls per hour.

(a) what is the probability distribution of X where X is the length of time between successive calls?

I'm saying it's an exponential distribution? But I'm not entirely sure...

(b) find the E(X) = miu and Var(X) = sigma^2

(c) find the probability that X is in the interval miu +/- 0.5sigma.

I'm thinking if it were an exponential distribution, then I integrate the density function from the lower to the upper limits?

(d) what is the probability that the time between successive calls to the ambulance service is longer than 30 minutes?

(e) What is the probability that there will be at least one emergency ambulance service call in the next 45 minutes, given that there was no call in the last 15 minutes?

2. Originally Posted by deathstarx
Hi, could someone help me out with this question?

Calls to an emergency ambulance service in a large city are modeled by a Poisson process with an average 3.2 calls per hour.

(a) what is the probability distribution of X where X is the length of time between successive calls?

I'm saying it's an exponential distribution? But I'm not entirely sure... Mr F says: Correct. In a poisson process the waiting times are exponentially distributed.

If the number of arrivals in a given time interval [0,t] follows the Poisson distribution, with mean = λt, then the lengths of the inter-arrival times follow the Exponential distribution, with mean 1 / λ.
(Quoted from Poisson distribution - Wikipedia, the free encyclopedia)

(b) find the E(X) = miu and Var(X) = sigma^2

(c) find the probability that X is in the interval miu +/- 0.5sigma.

I'm thinking if it were an exponential distribution, then I integrate the density function from the lower to the upper limits?

(d) what is the probability that the time between successive calls to the ambulance service is longer than 30 minutes?

(e) What is the probability that there will be at least one emergency ambulance service call in the next 45 minutes, given that there was no call in the last 15 minutes?

For (e), calculate $\displaystyle \Pr\left( \frac{1}{4} \leq X \leq 1 | X > \frac{1}{4} \right)$.