Probstat/notes/sample means and sample variances - ประวัติรุ่นแก้ไข

Jittat: /* Sample means */

2014-12-05T09:41:06Z

Sample means

2014-12-05T09:40:37Z

Sample means

2014-12-04T05:40:28Z

Why do we use normal distributions?

2014-12-03T13:18:26Z

2014-12-03T06:21:42Z

Examples

2014-12-03T06:21:06Z

Distribution of sample means

2014-12-03T06:20:57Z

Distribution of sample means

2014-12-03T06:20:12Z

2014-12-03T06:18:18Z

Why do we use normal distributions?

2014-12-03T06:14:27Z

Examples

@@ แถว 38: / แถว 38: @@
 <center>
-<math>Var(\bar{X}) = Var\left(\frac{X_1+X_2+\cdots+X_n}{n}\right) = \frac{1}{n^2}Var(X_1+X_2+\cdots+X_n) = \frac{1}{n^2}nVar(X) = \frac{\sigma^2}{n}.</math>
+<math>Var(\bar{X}) = Var\left(\frac{X_1+X_2+\cdots+X_n}{n}\right) = \frac{1}{n^2}Var(X_1+X_2+\cdots+X_n) = \frac{1}{n^2}\cdot n\cdot Var(X) = \frac{\sigma^2}{n}.</math>
 </center>

@@ แถว 35: / แถว 35: @@
 </center>
-and
+and since <math>X_1,X_2,\ldots,X_n</math> are independent, we have that
 <center>
-<math>Var(\bar{X}) = \frac{\sigma^2}{n}.</math>
+<math>Var(\bar{X}) = Var\left(\frac{X_1+X_2+\cdots+X_n}{n}\right) = \frac{1}{n^2}Var(X_1+X_2+\cdots+X_n) = \frac{1}{n^2}nVar(X) = \frac{\sigma^2}{n}.</math>
 </center>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 05:40, 4 ธันวาคม 2557
แถว 199:		แถว 199:

	== Why do we use normal distributions? ==		== Why do we use normal distributions? ==
−	~~: ''To be added~~...''	+	Normal random variables appear very often in our treatment of statistics. This is not just a coincidence. See [[probstat/notes/limit theorems\|limit theorems]].

@@ แถว 10: / แถว 10: @@
 <center>
-<math>\sigma^2=Var(X)=E[(X-\mu)^2]=E[X^2] = E[X]^2.</math>.
+<math>\sigma^2=Var(X)=E[(X-\mu)^2]=E[X^2] - E[X]^2.</math>
 </center>

@@ แถว 193: / แถว 193: @@
 which is roughly 1%.
 == Why do we use normal distributions? ==
 : ''To be added...''

@@ แถว 159: / แถว 159: @@
 {{กล่องเทา|
 '''Distribution of sample means of normal populations.'''
 If the population is normally distributed with mean <math>\mu</math> and variance <math>\sigma^2</math>, the distribution of <math>\bar{X}</math> is normal with mean <math>\mu</math> and variance <math>\sigma^2/n</math>.
 }}

@@ แถว 122: / แถว 122: @@
 </center>
-=== Summary: properties of sample means and sample variances ===
+=== Summary ===
 {{กล่องเทา|

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 06:18, 3 ธันวาคม 2557
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	== Why do we use normal distributions? ==		== Why do we use normal distributions? ==
		+	: ''To be added...''