Respuesta :
Answer: a) 0.333, b) 0.333 and c) 0.9986.
Step-by-step explanation:
Since we have given that
Average = 3 counts per minute = λ
a) What is the mean time between counts ?
Since it follows a Poisson Process, So,
[tex]E[x]=\dfrac{1}{\lambda}=\dfrac{1}{3}=0.333[/tex]
(b) What is the standard deviation between counts ?
[tex]\sigma=\dfrac{1}{3}=0.333[/tex]
(c) If it is an average of 3 counts per minute, find the value of such that .
If the average = 3 counts per minute.
Then, [tex]P(X<x)=0.95\\\\\int\limits^x_0 {\lambda e^{-\lambda t}} \, dt=0.95\\\\3\int\limits^x_0 {e^{-3t}} \, dt =0.95\\\\-e^{-3t}\mid ^{x}_0=0.95\\\\-3x=\ln (0.95)\\\\x=\dfrac{-0.05}{3}=0.9986[/tex]
Hence, a) 0.333, b) 0.333 and c) 0.9986.
You can use the mean and variance for Poisson distribution to get the needed information.
The answers are:
a) The mean time between counts is 0.333 minutes
b) The standard deviation between counts (in minutes) is 0.577 minutes
What are some of the properties of Poisson distribution?
Let X ~ Pois(λ)
Then we have:
E(X) = λ = Var(X)
Since standard deviation is square root (positive) of variance,
Thus,
Standard deviation of X = [tex]\sqrt{\lambda}[/tex]
Its probability function is given by
f(k; λ) = Pr(X = k) = [tex]\dfrac{\lambda^{k}e^{-\lambda}}{k!}[/tex]
Using the above facts and other facts to calculate the needed information
Since it is given that the log-ons follow Poisson distribution, thus, we have:
Counts per minute = C ~ Pois( λ = 3)
Time per count = T ~ Pois(λ = 1/3)
Mean time between counts = λ = 1/3 = 0.333 minutes approx.
Standard deviations of time between counts(in minutes) = [tex]\sqrt{\lambda} = \sqrt{\dfrac{1}{3}} \approx 0.577[/tex] minutes
Thus,
The answers are:
a) The mean time between counts is 0.333 minutes
b) The standard deviation between counts (in minutes) is 0.577 minutes
Learn more about Poisson distribution here:
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