# Central Limit Theorem

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A Central Limit Theorem is a probability theorem which states that for a set $\displaystyle{ X }$ of $\displaystyle{ n }$ number of independent and identically distributed random variable sample (each with expected value $\displaystyle{ \mu }$ and variance $\displaystyle{ \sigma^2 }$) of sufficiently large set size $\displaystyle{ n }$, the probability distribution of the sample mean $\displaystyle{ \bar X }$ of is approximately normal (with mean $\displaystyle{ \mu }$ and variance $\displaystyle{ {1}{n} \sigma^2 }$), and the sample total distribution is approximately normal with mean $\displaystyle{ n\mu }$, and variance $\displaystyle{ n\sigma^2 }$

• AKA: CLT.
• Context:
• Example(s):
• In an experiment of throwing of a die, let us say 1000 samples $\displaystyle{ S_1, S_2, S_3,\dots\, S_{1000} }$have been taken. Each sample $\displaystyle{ S_i }$ is of sample size $\displaystyle{ n=5 }$. That is if a die thrown five times and the output showed up as 1, 3, 3, 4, 2; then $\displaystyle{ S_1=[1,3,3,4,2] }$. Similarly in the next five throws the output showed up as 1, 1, 2, 6, 6;then $\displaystyle{ S_2=[1,1,2,6,6] }$ and so on. Now writing the samples and their respective means we get

$\displaystyle{ S_1=[1,3,3,4,2]; \mu_1=2.6 }$(mean of $\displaystyle{ S_1 }$)

$\displaystyle{ S_2=[1,1,2,6,6]; \mu_2=3.2 }$(mean of $\displaystyle{ S_2 }$)

$\displaystyle{ S_3=[1,6,5,2,4]; \mu_3=3.6 }$(mean of $\displaystyle{ S_3 }$)

$\displaystyle{ S_{1000}=[1,1,4,6,6]; \mu_{1000}=3.6 }$(mean of $\displaystyle{ S_{1000} }$)

By plotting all the sample means by keeping mean values ($\displaystyle{ \mu_i }$) along x-axis and their frequencies along y-axis it can be observed that the sample distribution looks some what normal. Then if the size of the samples increases from $\displaystyle{ n=5 }$ to $\displaystyle{ n=20 }$, the distribution will be more close to a normal distribution. If the sample size n increases to 100, then the sample distribution will be even more closer to normal distribution then before two cases. So when the sample size $\displaystyle{ n\to\infty }$ the sampling distribution becomes a perfect normal distribution (mean, median and mode all are same). This is what called Central Limit Theorem.

• Counter-Example(s):
• See: Statistical Independence, Random Variate, Probability Distribution, Identically Distributed, Weak Convergence of Measures, Independent And Identically Distributed Random Variables, Attractor.

## References

### 2014

1. David Williams, "Probability with martingales", Cambridge 1991/2008