Question

The time intervals between successive barges passing a certain point on a busy waterway have an exponential distribution with mean 8 minutes. a) Find the probability that the time interval between two successive barges is less than 5 minutes. b) Find a time interval 𝑡 such that we can be 95% sure that the time interval between two successive barges will be greater than 𝑡.

Answer 1

We have $\mu = 8,$ so $\lambda= 0.125.$

(a) The probability is

$$P(T < 5) =\displaystyle\int_{0}^{5}0.125e^{-0.125t}dt=-[e^{-0.125t}]\begin{matrix} 5 \\ 0 \end{matrix}$$ $$=-e^{-0.125(5)}+e^{-0.125(0)}=0.4647$$

(b) We require

$$P(T >t) =\displaystyle\int_{t}^{\infin}0.125e^{-0.125x}dx$$

$$=\lim\limits_{A\to \infin}[-e^{-0.125x}]\begin{matrix} A \\ t \end{matrix}=e^{-0.125t}=0.95$$

$$t=-8\ln0.95\ min$$

$$t\approx 0.4103\ min=24.62\ s$$