Q. 5.13 - A First Course in Probability

Question

At a certain bank, the amount of time that a customer spends being served by a teller is an exponential random variable with mean $5$ minutes. If there is a customer in service when you enter the bank, what is the probability that he or she will still be with the teller after an additional $4$ minutes?

Step-by-Step Solution

Answer

The necessary probability  of bank is around $0.45$.

Step :1 Random variables

Establish $X$ as a  variable quantity  that represents the number of  your  time spent at the bank teller. We're given the knowledge that $X~\mathrm{Expo}\left(\lambda \right)$ and $E X=5$ , implying that $\lambda =\frac{1}{5}$. Assume there's a customer at the teller once  we enter the bank. We are able to  leverage the memoryless property of the exponential distribution, which states that we can reset time and observe from the beginning at any time. Assume there's a  customer has only recently begun functioning  at the teller. The likelihood that he will remain unmoved

Step :2 Substitution

$$\begin{gathered} P(X>4)=1-P(X\le 4) \\ =1-\left(1-e^{-\frac{1}{5}\times 4}\right) \\ =e^{-\frac{4}{5}} \\ =0.45 \\ P(X>4)\approx 0.45 \end{gathered}$$