On average, 3 oil tankers arrive at a port each day, The time between arrivals follows an exponential distribution.

Question

(a) Find the probability that the next tanker does not arrive until at least two days from now.

(b) Find the probability that seven tankers arrive in one day.

for first question we have the scale parameter $\lambda = \frac13$

so we have to calculate $ P(T > 2) = \int_2^{ \infty} \frac13 \exp {(-\frac13x)}\,dx$

for second question I'm a bit lost, I feel like the notion of multiple random variables should be introduced but I don't know how.

an idea that popped in my head is that the sum of independent arrival times should be less than 24, the sum of exponentials give us a gamma and then it's just computation of an integral.

is my thinking correct ?

Answer 1

Let $T_{1},T_{2},\ldots$ be a sequence of independent exponential random variables with parameter $\lambda=1/3$. Let $S_{n}\equiv T_{1}+\cdots+T_{n}$. By the memoryless property of the exponential distribution, $S_{n}$ is the time at which the $n$-th tanker arrives. Note also that $S_{n}$ has a Gamma distribution with parameters $n$ and $\lambda$.