Central limit theorem by example - Revision history

Alexandk at 01:11, 17 December 2021

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← Older revision		Revision as of 01:11, 17 December 2021
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−	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:	+	The Central Limit Theorem 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>Y_1, Y_2,\dots ,Y_n, \dots </math> be a sequence of independent identically distributed random variables with mean <math>\mu </math> and variance <math>\sigma ^2 </math>. Let <math>\overline{Y}_n = (Y_1+Y_2+\dots +Y_n)/n </math> and		Let <math>Y_1, Y_2,\dots ,Y_n, \dots </math> be a sequence of independent identically distributed random variables with mean <math>\mu </math> and variance <math>\sigma ^2 </math>. Let <math>\overline{Y}_n = (Y_1+Y_2+\dots +Y_n)/n </math> and

← Older revision		Revision as of 01:10, 17 December 2021
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	(use induction or the fact that sum of independent identically distributed exponential random variables has Gamma distribution; the latter can be shown using moment-generating functions).		(use induction or the fact that sum of independent identically distributed exponential random variables has Gamma distribution; the latter can be shown using moment-generating functions).
		+

	'''Example: sums of uniformly distributed random variables'''		'''Example: sums of uniformly distributed random variables'''

@@ Line 131: / Line 131: @@
 [[File:uni2.png|alt=Density f2|thumb|Density of Y1+Y2|none]]
-<math> f_4(y) =
+<math> f_3(y) =
 \begin{cases}
 \frac{y^{2}}{2} & 0\le y \le 1 \\

@@ Line 143: / Line 143: @@
 [[File:uni3.png|alt=Density f3|thumb|Density of Y1+Y2+Y3|none]]
-<math> f_4(y) =
+<math> f_4(y)
 </math>
 [[File:uni4.png|alt=Density f4|thumb|Density of Y1+Y2+Y3+Y4|none]]

@@ Line 138: / Line 138: @@
 &\mbox{ otherwise}
 \end{cases}
 </math>

← Older revision		Revision as of 01:04, 17 December 2021
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	<math> f_3(y) =		<math> f_3(y) =
		+	\begin{cases} 0 & \text{for}\: x \leq 0 \\\frac{x^{2}}{2} & \text{for}\: x \leq 1 \\- x^{2} + 3 x - \frac{3}{2} & \text{for}\: x \leq 2 \\\frac{x^{2}}{2} - 3 x + \frac{9}{2} & \text{for}\: x \leq 3 \\0 & \text{otherwise} \end{cases}

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 <math> f_3(y) =
-\begin{cases} 0 & \text{for}\: x \geq 2 \vee x \leq 0 \\x & \text{for}\: x \leq 1 \\2 - x & \text{for}\: x \geq 1 \\1 & \text{otherwise} \end{cases}
 </math>