This is a Statistics problem. Is the sample mean from the Poisson distribution normally distributed? Example 5.21Let X1,..., X30 be independent Poisson random variables with rate 2. From our knowledge of thePoisson distribution, each X, has mean u = 2 and standard deviation o = v2. Assuming n = 30 isa large enough sample size, the Central Limit Theorem says thatX – 2Z =V2/V30will be approximately normal with mean 0 and standard deviation 1. Let us check this with asimulation.This is a little bit more complicated than our previous examples, but the idea is still the same. Wecreate an experiment which computes X and then transforms it by subtracting 2 and dividing byV2/V30.Here is a single experiment:Xbar <- mean (rpois (30, 2))(Xbar - 2) / (sgrt(2) / sgrt(30))## [1] 0.1290994Now, we replicate and plot:z <- replicate(10000, {Xbar <- mean ( rpois (30, 2))(Xbar - 2) / (sqrt (2) / sqrt(30))})plot(density(Z),main = "Standardized sum of 30 Poisson rvs", xlab = "Z"curve(dnorm(x), add = TRUE, col = "red")Standardized sum of 30 Poisson rvsFigure 5.4: Standardized sum of 30 Poisson random variables compared to a standard normal rv.In Figure 5.4 we see very close agreement between the simulated density of Z and the standardnormal density curve.Density0.0 0.1 0.2 0.3 0.4TIT

A Poisson distribution is the probability of the number of events that occur in a given interval…

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This is a Statistics problem.

Is the sample mean from the Poisson distribution normally distributed?

Example 5.21
Let X1,..., X30 be independent Poisson random variables with rate 2. From our knowledge of the
Poisson distribution, each X, has mean u = 2 and standard deviation o = v2. Assuming n = 30 is
a large enough sample size, the Central Limit Theorem says that
X – 2
Z =
V2/V30
will be approximately normal with mean 0 and standard deviation 1. Let us check this with a
simulation.
This is a little bit more complicated than our previous examples, but the idea is still the same. We
create an experiment which computes X and then transforms it by subtracting 2 and dividing by
V2/V30.
Here is a single experiment:
Xbar <- mean (rpois (30, 2))
(Xbar - 2) / (sgrt(2) / sgrt(30))
## [1] 0.1290994
Now, we replicate and plot:
z <- replicate(10000, {
Xbar <- mean ( rpois (30, 2))
(Xbar - 2) / (sqrt (2) / sqrt(30))
})
plot(density(Z),
main = "Standardized sum of 30 Poisson rvs", xlab = "Z"
curve(dnorm(x), add = TRUE, col = "red")
Standardized sum of 30 Poisson rvs
Figure 5.4: Standardized sum of 30 Poisson random variables compared to a standard normal rv.
In Figure 5.4 we see very close agreement between the simulated density of Z and the standard
normal density curve.
Density
0.0 0.1 0.2 0.3 0.4
TIT

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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