Question

3-141. The time between the arrival of e-mail messages at your computer is exponentially distributed with a mean of 2 hours. (a) What is the probability that you do not receive a message during a 2-hour period?

Answer 1

The probability that there was no messages received during a 2-hour period is 0.3679.

Explanation:

Let the random variable X = time between the arrival of e-mail messages.

The random variable
`X\sim Exp(\lambda)`

The probability distribution function of exponential distribution is:

`f(x)=\left \{ {{\lambda e^(-\lambda x);\ x>0} \atop {0};\ otherwise} \right.`

The mean of the distribution is, Mean = 2.

The value of λ is:

`\lambda=(1)/(Mean) =(1)/(2)=0.50`

Compute the probability that there was no messages received during a 2-hour period as follows:

`P(X>2)=1-P(X\leq 2)\\=1-\int\limits^(2)_(0) {\lambda e^(-\lambda x)} \, dx \\=1-\lambda[(e^(-\lambda x))/(-\lambda) ]^(2)_(0)\\=1-[1-e^{-(x)/(2) }]^(2)_(0)\\=1-[1-e^{-(2)/(2)}]\\=e^(-1)\\=0.3679`

Thus, the probability that there was no messages received during a 2-hour period is 0.3679.