Suppose that the log-ons to a computer network follow a Poisson process with an average of 3 counts per minute.

(a) What is the mean time between counts (in minutes)? (Round yours answers to 3 decimal places.)

(b) What is the standard deviation between counts (in minutes)? (Round yours answers to 3 decimal places.)

(c) If it is an average of 3 counts per minute, find the value of x such that P(X < x) = 0.95 (Round yours answers to 4 decimal places.)

a) 20 second

b) 20 second

c) 1 minute

Step-by-step explanation:

Let X be the exponential random variable arising from the Poisson process with the rate λ = 3 counts per minute. Its cdf is then given by:  

F(x) = I - e^-3x, x >= 0  

Calculate the mean and the standard deviation of the random variable X as follows:  

Е(X) =I/λ =I/3=20 second

std (x) =√1/λ^2=1/3=20 second

For the part c), write down the equation:

0.95 = P(X < x) = F(x) = I - e^-3x

and solve the equation for x to obtain:

x = -1/3 In 0.05 = 0.9985 ≅  1 minute

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