Probstat/notes/limit theorems - ประวัติรุ่นแก้ไข

Jittat: /* Law of large numbers */

2014-12-08T01:29:05Z

Law of large numbers

Jittat: /* Central limit theorems */

2014-12-08T01:28:02Z

Central limit theorems

Jittat เมื่อ 01:27, 8 ธันวาคม 2557

2014-12-08T01:27:39Z

Jittat: /* Law of large numbers */

2014-12-08T01:24:03Z

Law of large numbers

Jittat: /* Law of large numbers */

2014-12-08T01:23:38Z

Law of large numbers

Jittat เมื่อ 01:23, 8 ธันวาคม 2557

2014-12-08T01:23:20Z

Jittat เมื่อ 01:16, 8 ธันวาคม 2557

2014-12-08T01:16:06Z

Jittat เมื่อ 01:15, 8 ธันวาคม 2557

2014-12-08T01:15:26Z

Jittat เมื่อ 01:08, 8 ธันวาคม 2557

2014-12-08T01:08:59Z

Jittat เมื่อ 01:08, 8 ธันวาคม 2557

2014-12-08T01:08:42Z

@@ แถว 12: / แถว 12: @@
 == Law of large numbers ==
 : ''See also [http://en.wikipedia.org/wiki/Law_of_large_numbers the wikipedia article]''.
 {{กล่องเทา|
@@ แถว 24: / แถว 26: @@
 as <math>n\rightarrow\infty</math>.
 }}
 == Central limit theorems ==

@@ แถว 34: / แถว 34: @@
 <center>
-<math>P\left\{\frac{X_1+X_2+\cdots+X_n-n\mu}{\sigma\sqrt{n}} \leq a \right\} \rightarrow \frac{1}{\sqrt{2\pi}}\int_{-infty}^a e^{-x^2/2} dx</math>
+<math>P\left\{\frac{X_1+X_2+\cdots+X_n-n\mu}{\sigma\sqrt{n}} \leq a \right\} \rightarrow \frac{1}{\sqrt{2\pi}}\int_{-\infty}^a e^{-x^2/2} dx</math>
 </center>

@@ แถว 27: / แถว 27: @@
 == Central limit theorems ==
 : ''See also [http://en.wikipedia.org/wiki/Central_limit_theorem the wikipedia article]''.
 == Links ==

@@ แถว 19: / แถว 19: @@
 <center>
-<math>P\left\{|\frac{X_1+X_2+\cdots+X_n}{n} - \mu| > \epsilon \right\} \rightarrow 0</math>
+<math>P\left\{\left|\frac{X_1+X_2+\cdots+X_n}{n} - \mu\right| > \epsilon \right\} \rightarrow 0</math>
 </center>

@@ แถว 14: / แถว 14: @@
 {{กล่องเทา|
-(The week law of large numbers)  We have random variables <math>X_1,X_2,\ldots, X_n</math> which are independent identically distributed with mean <math>\mu</math> and variance <math>\sigma^2</math>.  For any <math>\epsilon > 0</math>,
+'''The week law of large numbers'''
 <center>

@@ แถว 3: / แถว 3: @@
 In this section, we shall briefly describe one of the reasons why normal distributions appears very often in statistics.  This also justifies our assumptions in many applications that the populations we are working on are normally distributed.
-There are usually two types of limit results, each with many specific versions under many assumptions.  They are referred to as:
+Limit theorems, important results in probability theory, consider situations when the size of the sample is very large.  There are usually two types of limit results, each with many specific versions under many assumptions.  They are referred to as:
 * Laws of large numbers --- discuss the fact that as we increase the size of the sample, the sample mean converges to the population mean.
 * Central limit theorems -- discuss the fact that the distributions of the sample mean converges to a normal distribution.
-'''Settings:''' Here we describe the settings of both results.
+'''Settings:''' Here we describe the settings of both results.  We have random variables <math>X_1,X_2,\ldots, X_n</math> which are independent identically distributed with mean <math>\mu</math> and variance <math>\sigma^2</math>.  (This is true when we take a sample <math>X_1,X_2,\ldots,X_n</math> of size <math>n</math> from a population whose mean is <math>\mu</math> and variance <math>\sigma^2</math>.)
 == Law of large numbers ==
 : ''See also [http://en.wikipedia.org/wiki/Law_of_large_numbers the wikipedia article]''.
 == Central limit theorems ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:15, 8 ธันวาคม 2557
แถว 2:		แถว 2:

	In this section, we shall briefly describe one of the reasons why normal distributions appears very often in statistics. This also justifies our assumptions in many applications that the populations we are working on are normally distributed.		In this section, we shall briefly describe one of the reasons why normal distributions appears very often in statistics. This also justifies our assumptions in many applications that the populations we are working on are normally distributed.
		+
		+	There are usually two types of limit results, each with many specific versions under many assumptions. They are referred to as:
		+
		+	* Laws of large numbers --- discuss the fact that as we increase the size of the sample, the sample mean converges to the population mean.
		+	* Central limit theorems -- discuss the fact that the distributions of the sample mean converges to a normal distribution.

	== Law of large numbers ==		== Law of large numbers ==