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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MATLAB: An Introduction with Applications

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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

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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