Question

Let X be the amount of time (in minutes) a USPS clerk spends with a customer. It is known that X follows an exponential distribution and USPS clerks take care of 38.1 percent of customers less than two minutes. Find P(0.5 < X < 3).

Answer 1

Answer: 0.4

Explanation:

Given: Cumulative exponential distribution:

`P(x<k)=1-e^{(-x)/(k)}`

As per given,

`P(x<2)=1-e^{(-2)/(k)}=0.381\\\\\Rightarrow\ e^{(-2)/(k)}=1-0.381\\\\\Rightarrow\ e^{(-2)/(k)}=0.619\\\\\Rightarrow\ {(-2)/(k)}=\ln(0.619) \ \ \ [\text{Taking natural log on both sides }]\\\\\Rightarrow\ {(-2)/(k)}=-0.47965\\\\\Rightarrow\ k=(2)/(0.47965)\\\\\Rightarrow\ k=4.17`

`P(0.5 < X < 3) \\\\=P(X<3)-P(X<0.5) \\\\=(1-e^{-(3)/(4.17)})-(1-e^{-(0.5)/(4.17)}) \\\\=e^{-(0.5)/(4.17)}-e^{-(3)/(4.17)}\\\\= 0.4`

Hence, the required probability = 0.4