Question

Wingard Credit Union is also interested in understanding how long customers spend in front of the ATMs. Customer service times follow an Exponential distribution, with an average customer taking 1.5 minutes to complete a transaction.

Calculate the probability that a customer will take less than half a minute. Additionally, calculate the probability that a customer will take more than 3 minutes.

Answer 1

Answer:

The probability that a customer will take less than half a minute to complete a transaction is 0.2835.

The probability that a customer will take more than 3 minutes to complete a transaction is 0.1353.

Step-by-step explanation:

Let X = service time of customers at an ATM to complete a transaction.

The average service time is, β = 1.5 minutes.

The random variable X follows an Exponential distribution with parameter λ = 1/β.

The probability density function of X is:

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

(1)

Compute the probability that a customer will take less than half a minute to complete a transaction as follows:

$P(X<0.5)=\int\limits^{0.5}_{0} {\frac{1}{1.5} e^{-\frac{x}{1.5}}} \, dx \\=\frac{1}{1.5}\int\limits^{0.5}_{0} { e^{-\frac{x}{1.5}}} \, dx\\=\frac{1}{1.5}|\frac{e^{-\frac{x}{1.5}}}{-\frac{x}{1.5}}|^{0.5}_{0}\\=-e^{-\frac{0.5}{1.5}}+e^{-\frac{0}{1.5}}\\=1-0.7165\\=0.2835$

Thus, the probability that a customer will take less than half a minute to complete a transaction is 0.2835.

(2)

Compute the probability that a customer will take more than 3 minutes to complete a transaction as follows:

$P(X>3)=\int\limits^{\infty}_{3} {\frac{1}{1.5} e^{-\frac{x}{1.5}}} \, dx \\=\frac{1}{1.5}\int\limits^{\infty}_{3} { e^{-\frac{x}{1.5}}} \, dx\\=\frac{1}{1.5}|\frac{e^{-\frac{x}{1.5}}}{-\frac{x}{1.5}}|^{\infty}_{3}\\=-e^{-\frac{\infty}{1.5}}+e^{-\frac{3}{1.5}}\\=0+0.1353\\=0.1353$

Thus, the probability that a customer will take more than 3 minutes to complete a transaction is 0.1353.