Question

The caller times at a customer service center has an exponential distribution with an average of 22 seconds. Find the probability that a randomly selected call time will be less than 30 seconds? (Round to 4 decimal places.)

Answer 1

Answer:

The probability that a randomly selected call time will be less than 30 seconds is 0.7443.

Step-by-step explanation:

We are given that the caller times at a customer service center has an exponential distribution with an average of 22 seconds.

Let X = caller times at a customer service center

The probability distribution (pdf) of the exponential distribution is given by;

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

Here, $\lambda$ = exponential parameter

Now, the mean of the exponential distribution is given by;

Mean =  $\frac{1}{\lambda}$  

So,  $22=\frac{1}{\lambda}$  ⇒ $\lambda=\frac{1}{22}$

SO, X ~ Exp($\lambda=\frac{1}{22}$)  

To find the given probability we will use cumulative distribution function (cdf) of the exponential distribution, i.e;

    $P(X\leq x) = 1 - e^{-\lambda x}$  ; x > 0

Now, the probability that a randomly selected call time will be less than 30 seconds is given by = P(X < 30 seconds)

        P(X < 30)  =  $1 - e^{-\frac{1}{22} \times 30}$

                         =  1 - 0.2557

                         =  0.7443