Question

The time between arrivals of customers at the drive-up window of a bank follows an exponential probability distribution with a Mean of 13. What is the probability that the arrival time between customers will be 12 or less? Report your answers in decimals, using 4 decimals.

Answer 1

The probability that the arrival time between customers will be 12 or less is 0.6027.

Explanation:

We are given that the time between arrivals of customers at the drive-up window of a bank follows an exponential probability distribution with a Mean of 13.

Let X = time between arrivals of customers

The probability distribution for exponential distribution is given by;

`f(x) = \lambda e^(-\lambda x) ; x >0`

where,
`\lambda` = parameter of this distribution or the arrival rate

Since, the mean of exponential distribution = E(X) =
`(1)/(\lambda)`

So, 13 =
`(1)/(\lambda)` ,
`\lambda=(1)/(13)`

So, X ~ Exp(
`\lambda=(1)/(13)` )

Now, to find the less than or greater than probabilities in exponential distribution we use the Cumulative distribution function of exponential function, i.e.;

`F(x) = P(X \leq x) = 1 - e^(-\lambda x) ; x >0`

So, probability that the arrival time between customers will be 12 or less is given by = P(X
`\leq` 12)

P(X
`\leq` 12) =
`1 - e^(-\lambda x)`

=
`1 - e^{-(1)/(13) * 12}`

=
`1 - e^{-(12)/(13) }` = 0.6027

Therefore, probability that the arrival time between customers will be 12 or less is 0.6027.