Question

Suppose that the amount of time one spends in a bank is exponentially distributed with mean 10 minutes. What is the probability that a customer will spend more than 15 minutes in the bank? What is the probability that a customer will spend more than 15 minutes in the bank given that he is still in the bank after 10 minutes?

Answer 1

Let $X=$ the amount of time a customer spends in a bank: $X\sim Exp(\lambda).$

$$f(x)=\lambda e^{-\lambda x}, x>0$$

$\lambda=\dfrac{1}{\mu}=\dfrac{1}{10}=0.1$

$$P(X>15)=\displaystyle\int_{15}^{\infin}f(x)dx$$

$$=\lim\limits_{A\to\infin}\displaystyle\int_{15}^{A}0.1 e^{-0.1 x}dx$$

$$=\lim\limits_{A\to\infin}[-e^{0.1x}]\begin{matrix} A \\ 15 \end{matrix}=-0+e^{-0.1(15)}$$

$$=e^{-1.5}\approx0.22313$$

An exponential distribution has a memory-less property, i.e the future probabilities are not affected by any past data.

$$P(X>15|X>10)=P(X>10+5|X>10)$$

$$=P(X>5)=\lim\limits_{A\to\infin}\displaystyle\int_{5}^{A}0.1 e^{-0.1 x}dx$$

$$=\lim\limits_{A\to\infin}[-e^{0.1x}]\begin{matrix} A \\ 5 \end{matrix}=-0+e^{-0.1(5)}$$

$$=e^{-0.5}\approx0.60653$$