Difference between revisions of "Central limit theorem by example"
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| + | While the proof of this theorem is often beyond the scope of introductory undergraduate probability and statistics courses, there are several "convincing" examples that make the statement of the theorem very plausible. Below we provide two such example. | ||
| − | + | '''Bernoulli trials and Binomial distribution''' | |
| − | \ | + | Let <math> Y_1,Y_2,\dots , Y_n,\dots <\math> be random variables representing Bernoulli trials, i.e. <math> P(Y_n=1)=p </math> and <math> P(Y_n=0)=1-p </math> for all <math> n </math>. Then <math> X_n= Y_1+Y_2+\dots +Y_n </math> has Binomial distributions with parameters <math> p </math> and <math> n </math>. A concrete examples here would be rolling a die repeatedly, with success being, say, rolling a 1. For smaller <math> n </math> (e.g. <math> n= 10 </math>) the Binomial histogram is not symmetric. However, for larger <math> n </math> the histogram of the distribution of <math> X_n </math> resembles the normal density curve. |
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| + | '''Iterated convolution''' | ||
\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 | \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 | ||
Revision as of 11:41, 22 November 2021
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].
While the proof of this theorem is often beyond the scope of introductory undergraduate probability and statistics courses, there are several "convincing" examples that make the statement of the theorem very plausible. Below we provide two such example.
Bernoulli trials and Binomial distribution
Let [math]\displaystyle{ Y_1,Y_2,\dots , Y_n,\dots \lt \math\gt be random variables representing Bernoulli trials, i.e. \lt math\gt P(Y_n=1)=p }[/math] and [math]\displaystyle{ P(Y_n=0)=1-p }[/math] for all [math]\displaystyle{ n }[/math]. Then [math]\displaystyle{ X_n= Y_1+Y_2+\dots +Y_n }[/math] has Binomial distributions with parameters [math]\displaystyle{ p }[/math] and [math]\displaystyle{ n }[/math]. A concrete examples here would be rolling a die repeatedly, with success being, say, rolling a 1. For smaller [math]\displaystyle{ n }[/math] (e.g. [math]\displaystyle{ n= 10 }[/math]) the Binomial histogram is not symmetric. However, for larger [math]\displaystyle{ n }[/math] the histogram of the distribution of [math]\displaystyle{ X_n }[/math] resembles the normal density curve.
Iterated convolution
\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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