Respuesta :
Answer:
The probability that there was no messages received during a 2-hour period is 0.3679.
Step-by-step explanation:
Let the random variable X = time between the arrival of e-mail messages.
The random variable [tex]X\sim Exp(\lambda)[/tex]
The probability distribution function of exponential distribution is:
[tex]f(x)=\left \{ {{\lambda e^{-\lambda x};\ x>0} \atop {0};\ otherwise} \right.[/tex]
The mean of the distribution is, Mean = 2.
The value of λ is:
[tex]\lambda=\frac{1}{Mean} =\frac{1}{2}=0.50[/tex]
Compute the probability that there was no messages received during a 2-hour period as follows:
[tex]P(X>2)=1-P(X\leq 2)\\=1-\int\limits^{2}_{0} {\lambda e^{-\lambda x}} \, dx \\=1-\lambda[\frac{e^{-\lambda x}}{-\lambda} ]^{2}_{0}\\=1-[1-e^{-\frac{x}{2} }]^{2}_{0}\\=1-[1-e^{-\frac{2}{2}}]\\=e^{-1}\\=0.3679[/tex]
Thus, the probability that there was no messages received during a 2-hour period is 0.3679.
