Simulation of the Law of the Large Numbers

Simulation is important to develop our intuitive understanding. So when the difficult content appears, try considering a sample. If there’s no handy sample, creating a sample with R is very efficient, simulation in other words.

When we want to understand the law of the large number, we can also generate it virtually. Here, I will show you the simulation. Consider the function which takes $$x \in [0, 3)$$ and the value is in $$[0, 1)$$, such as $$g(x) = \frac{1}{4} x(3-x)(x-0.75)^2$$. ($$g(x)$$ can’t be a density function for the random variable, because $$\int _0 ^1 g(x) dx \neq 1$$. It is the probability to admit $$x$$ value generated by randomly. It is for creating trial data.)

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Similarity among Binomial, Poisson and Normal Distributions

Binomial Distribution $$\textrm{Binomial}(m, p)$$ is similar to Normal and Poisson distributions.

Especially $$n$$ is large, it can be approximated as a normal distribution, according to the central limit theorem.

Let’s see the similarity with graphics.

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