Central limit theorem by example
The Central Limit Theorem (CLT for short) is one of the most fundamental results in Probability and Statistics, that provides numerous applications and, to some extent, "explains" ubiquity of normal distribution. Below is one of the versions of this theorem:
Let [math]\displaystyle{ Y_1, Y_2,\dots ,Y_n, \dots }[/math] be a sequence of independent identically distributed random variables with mean [math]\displaystyle{ \mu }[/math] and variance [math]\displaystyle{ \sigma ^2 }[/math]. Let [math]\displaystyle{ \overline{Y}_n = (Y_1+Y_2+\dots +Y_n)/n }[/math] and
[math]\displaystyle{ X_n = \frac{\overline{Y}_n-\mu}{\sigma/\sqrt{n}}. }[/math]
Then [math]\displaystyle{ \{ X_n\}_{n=1}^\infty }[/math] converges in distribution to the standard normal random variable, i.e.
[math]\displaystyle{ \lim _{n\to\infty} P(X_n\le x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt }[/math]
for all [math]\displaystyle{ x }[/math].
\item {\bf Example.} Let $Y_1,Y_2,\dots , Y_n,\dots$ be random variables representing Bernoulli trials, i.e.\\ $P(Y_n=1)=p$ and $P(Y_n=0)=1-p$ for all $n$. Then $X_n= Y_1+Y_2+\dots +Y_n$ has Binomial distributions with parameters $p$ and $n$. Note that for large $n$ the histogram of the distribution of $X_n$ resembles the normal density curve.
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\item {\bf Definition.} We say that a sequence of random variables $\{ X_n\} _{n=1}^\infty$ {\bf converges in distribution} to a random variable $X$ if $$\lim_{n\to\infty} P(X_n\le x) = F(x) = P(X\le x)$$ for all $x$ at which $F(x)$ is continuous.
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