Answer:
The length of the interval during which no messages arrive is 90 seconds long.
Step-by-step explanation:
Let X = number of messages arriving on a computer server in an hour.
The mean rate of the arrival of messages is, λ = 11/ hour.
The random variable X follows a Poisson distribution with parameter λ = 11.
The probability mass function of X is:
[tex]P(X=x)=\frac{e^{-11}11^{x}}{x!};\ x=0,1,2,3....[/tex]
It is provided that in t hours the probability of receiving 0 messages is,
P (X = 0) = 0.76
Compute the value of t as follows:
[tex]P(X=0)=\frac{e^{-11\times t}(11\times t)^{0}}{0!}\\0.76=e^{-11\times t}\\\ln(0.76)=-11\times t\\t=-\frac{\ln(0.76)}{11} \\=0.025\ hours\\\approx90\ seconds[/tex]
Thus, the length of the interval during which no messages arrive is 90 seconds long.
