Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is minutes. Find the median waiting time.
A) minutes
B) minutes
C) minutes
D) minutes
E) minutes

Question

Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is  minutes. Find the median waiting time.
A) minutes 
B) minutes 
C) minutes 
D) minutes 
E) minutes

Quizplus · Accepted Answer

The median waiting time is the value of 𝑥 for which the probability of waiting less than or equal to 𝑥 is 0.5.

Let 𝜆 be the rate parameter of the exponential distribution, then the probability density function is

𝑓(𝑡)=𝜆𝑒^(−𝜆𝑡)

The cumulative distribution function is

𝐹(𝑡)=∫_(0)^𝑡 𝜆𝑒^(−𝜆𝑥) 𝑑𝑥= 1−𝑒^(−𝜆𝑡)

Setting this to 0.5 and solving for 𝑡, we get

0.5= 1−𝑒^(−𝜆𝑡)

0.5= 𝑒^(−𝜆𝑡)

Taking the natural logarithm of both sides, we have

ln(0.5)=−𝜆𝑡

𝑡=−ln(0.5)/𝜆

The rate parameter 𝜆 is the reciprocal of the mean waiting time, so 𝜆=1/11eaa3e5_5531_1711_9f8f_3988b534741b_TB5971_11=9.587 × 10^-10

Plugging this into the formula for 𝑡, we get

𝑡=−ln(0.5)/𝜆≈7.225 minutes

This value is closest to option B: 11eaa3e5_5531_3e23_9f8f_0dbb594f7074_TB5971_11 minutes.

Suppose the Average Waiting Time for a Customer's Call to Be