Probstat/notes/balls and bins - ประวัติรุ่นแก้ไข

Jittat: /* The Linearity of Expectation */

2014-09-18T07:38:24Z

The Linearity of Expectation

2014-09-18T07:37:52Z

The Linearity of Expectation

2014-09-18T07:35:17Z

Fullest bins

2014-09-18T07:34:17Z

Using the linearity of expectation

2014-09-18T07:31:50Z

Using the linearity of expectation

2014-09-18T07:21:53Z

Using the linearity of expectation

2014-09-18T07:21:28Z

Using the linearity of expectation

2014-09-18T07:16:13Z

Using the linearity of expectation

2014-09-18T07:01:35Z

Using the linearity of expectation

2014-09-18T06:28:38Z

Using the linearity of expectation

@@ แถว 28: / แถว 28: @@
 One of the reason why expectations are rather easy to work with is that they have a certain linear property stated as follows.
-{{กล่องฟ้า|'''Linearity of Expectation.''' For random variables ''X'' and ''Y'', we have that
+{{กล่องฟ้า|'''Linearity of Expectation.'''
 <center><math>\mathrm{E}[X+Y] = \mathrm{E}[X] + \mathrm{E}[Y].</math></center>

@@ แถว 30: / แถว 30: @@
 {{กล่องฟ้า|'''Linearity of Expectation.''' For random variables ''X'' and ''Y'', we have that
-<center><math>\mathrm{E}[X+Y] = \mathrm{E}[X] + \mathrm{E}[Y].</math></center>}}
+<center><math>\mathrm{E}[X+Y] = \mathrm{E}[X] + \mathrm{E}[Y].</math></center>
 This property is always true even when they are dependent.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 07:35, 18 กันยายน 2557
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	== Fullest bins ==		== Fullest bins ==
		+	In this section, we will try to give a guarantee on the number of balls in the fullest bin.
		+
		+	=== Alternative analysis ===

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 07:34, 18 กันยายน 2557
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	Note that as ''n'' gets close to infinity, <math>\left(1 - 1/n\right)^n</math> is close to <math>1/e</math>. Thus, <math>\mathrm{E}[X]\approx n/e</math>, or there are about 36.7% empty bins.		Note that as ''n'' gets close to infinity, <math>\left(1 - 1/n\right)^n</math> is close to <math>1/e</math>. Thus, <math>\mathrm{E}[X]\approx n/e</math>, or there are about 36.7% empty bins.
		+
		+	{{กล่องเทา\|'''Remark:''' For an indicator random variable ''A'', <math>\mathrm{E}[A] = P\{A = 1\}</math>. }}

	== Fullest bins ==		== Fullest bins ==

@@ แถว 74: / แถว 74: @@
 </math>
 </center>
 Note that as ''n'' gets close to infinity, <math>\left(1 - 1/n\right)^n</math> is close to <math>1/e</math>.  Thus, <math>\mathrm{E}[X]\approx n/e</math>, or there are about 36.7% empty bins.
 == Fullest bins ==

@@ แถว 65: / แถว 65: @@
 This implies that
 <center><math>
-\begin{equation}
+\begin{array}{rcl}
-X = 10
+\mathrm{E}[X] &=& \mathrm{E}\left[\sum_{i=1}^n X_i\right] \\
-\end{equation}
+&=& \sum_{i=1}^n \mathrm{E}[X_i]\\
 </math>
 </center>
 == Fullest bins ==

@@ แถว 56: / แถว 56: @@
 Therefore, if we can find <math>\mathrm{E}[X_i]</math> then we are done.  So let's consider indicator random variable ''X<sub>i</sub>''.  From the definition, we have that
-<center><math>\mathrm{E}[X_i] = 0\cdot P\{X_i = 0\} + 1\cdot P\{X_i = 1\}=P\{X_i = 1 \}</math>.</center>
+<center><math>\mathrm{E}[X_i] = 0\cdot P\{X_i = 0\} + 1\cdot P\{X_i = 1\}</math>,</center>
 == Fullest bins ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 07:01, 18 กันยายน 2557
แถว 53:		แถว 53:

	<center><math>\mathrm{E}\left[\sum_{i=1}^n X_i\right] = \sum_{i=1}^n \mathrm{E}[X_i]</math>.</center>		<center><math>\mathrm{E}\left[\sum_{i=1}^n X_i\right] = \sum_{i=1}^n \mathrm{E}[X_i]</math>.</center>
		+
		+	Therefore, if we can find <math>\mathrm{E}[X_i]</math> then we are done. So let's consider indicator random variable ''X<sub>i</sub>''. From the definition, we have that
		+
		+	<center><math>\mathrm{E}[X_i] = 0\cdot P\{X_i = 0\} + 1\cdot P\{X_i = 1\}=P\{X_i = 1 \}</math>.</center>

	== Fullest bins ==		== Fullest bins ==

@@ แถว 45: / แถว 45: @@
 (You should think for a few minutes so that you understand this.)
 Note that we use each random variable ''X<sub>i</sub>'' to indicate if each bin is empty.  This type of random variables is generally referred to as '''indicator random variables'''.
 == Fullest bins ==