Probstat/notes/basic - ประวัติรุ่นแก้ไข

Jittat: /* Bayes' formula */

2014-09-18T03:58:29Z

Bayes' formula

2014-09-18T03:58:03Z

Independence

2014-09-18T03:57:28Z

Conditional probabilities

2014-09-18T03:55:10Z

Probability axioms

2014-09-13T04:28:12Z

Bayes' formula

2014-09-13T04:27:46Z

Bayes' formula

2014-09-13T04:27:00Z

Bayes' formula

2014-09-13T04:25:08Z

Useful properties

2014-09-13T04:24:31Z

Bayes' formula

2014-09-13T04:23:15Z

Useful properties

@@ แถว 62: / แถว 62: @@
 For any events ''A'' and ''B'', if we know <math>P(A)</math>, <math>P(B|A)</math> and <math>P(B|A^{c})</math>, we can find the "inverse" probability
 <math>P(A|B) = \frac{P(AB)}{P(B)} = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|A^c)P(A^c)} = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|A^c)(1 - P(A))}</math>.
 === Example ===

@@ แถว 44: / แถว 44: @@
 == Independence ==
-Events ''A'' and ''B'' are independent when the knowledge that ''A'' has occurs does not change the probability that ''B'' occurs.  Formally, we say that ''A'' and ''B'' are '''independent''' if and only if
+Events ''A'' and ''B'' are independent when the knowledge that ''A'' has occurs does not change the probability that ''B'' occurs.
-<math>P(AB)=P(A)P(B)</math>
+{{กล่องฟ้า|Formally, we say that ''A'' and ''B'' are '''independent''' if and only if
 This implies that if <math>P(A)\neq 0</math>, <math>P(B)=P(B|A)</math>.

@@ แถว 36: / แถว 36: @@
 == Conditional probabilities ==
-Suppose that we know that event ''B'' has occurred.  We denote by <math>P(A|B)</math> the probability that event ''A'' occurs given that event ''B'' has occurred.  Intuitively under this condition, we know that if ''A'' is going to occur, it must occur with ''B'', i.e., event ''AB'' must occur.  When <math>P(B)\neq 0</math>, we define
+Suppose that we know that event ''B'' has occurred.  We denote by <math>P(A|B)</math> the probability that event ''A'' occurs given that event ''B'' has occurred.  Intuitively under this condition, we know that if ''A'' is going to occur, it must occur with ''B'', i.e., event ''AB'' must occur.
-<math>P(A|B)=\frac{P(AB)}{P(B)}</math>.
+{{กล่องฟ้า|When <math>P(B)\neq 0</math>, we define the probability of event ''A'' given that event ''B'' has occurred as
 == Independence ==

@@ แถว 11: / แถว 11: @@
 We would like to assign probabilities to events.  Let ''S'' denote the sample space.  Formally a function ''P'' is a probability function if it satisfies the follow 3 axioms.
-'''1:''' For any event ''E'', <math>0\leq P(E)\leq 1</math>.
+{{กล่องฟ้า|'''Axiom 1:''' For any event ''E'', <math>0\leq P(E)\leq 1</math>.}}
-'''2:''' <math>P(S)=1</math>.
+{{กล่องฟ้า|'''Axiom 2:''' <math>P(S)=1</math>.}}
-'''3:''' For any countable sequence of mutually exclusive events <math>E_1,E_2,\ldots</math>, we have that
+{{กล่องฟ้า|'''Axiom 3:''' For any countable sequence of mutually exclusive events <math>E_1,E_2,\ldots</math>, we have that
-<math>P(\bigcup_i E_i) = \sum_i P(E_i)</math>.
+<center><math>P(\bigcup_i E_i) = \sum_i P(E_i)</math>.</center>
 == Useful properties ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 04:28, 13 กันยายน 2557
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	<math>P(A\|B) = \frac{P(AB)}{P(B)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)P(A^c)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)(1 - P(A))}</math>.		<math>P(A\|B) = \frac{P(AB)}{P(B)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)P(A^c)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)(1 - P(A))}</math>.
		+
		+	=== Example ===

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 04:27, 13 กันยายน 2557
แถว 55:		แถว 55:
	For any events ''A'' and ''B'', if we know <math>P(A)</math>, <math>P(B\|A)</math> and <math>P(B\|A^{c})</math>, we can find the "inverse" probability		For any events ''A'' and ''B'', if we know <math>P(A)</math>, <math>P(B\|A)</math> and <math>P(B\|A^{c})</math>, we can find the "inverse" probability

−	<math>P(A\|B) = \frac{P(AB)}{P(B)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)(1- P(A^c))}</math>.	+	<math>P(A\|B) = \frac{P(AB)}{P(B)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)P(A^c)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)(1 - P(A))}</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 04:27, 13 กันยายน 2557
แถว 53:		แถว 53:

	== Bayes' formula ==		== Bayes' formula ==
−	For any events ''A'' and ''B'', if we know <math>P(B\|A)</math> and <math>P(B\|A^{c})</math>.	+	For any events ''A'' and ''B'', if we know <math>P(A)</math>, <math>P(B\|A)</math> and <math>P(B\|A^{c})</math>, we can find the "inverse" probability
		+
		+	<math>P(A\|B) = \frac{P(AB)}{P(B)} = \frac{P(B\|A)P(A)}{P(B\|A)P(A) + P(B\|A^c)(1- P(A^c))}</math>.

@@ แถว 24: / แถว 24: @@
 '''1:''' <math>P(\empty) = 0</math>.
-'''2:''' For events ''A'' and ''B'', <math>P(A\cup B)\geq P(A)</math>.
+'''2:''' <math>P(A^c) = 1 - P(A)</math>.
-'''3:''' For events ''A'' and ''B'', <math>P(A\cap B)\leq P(A)</math>.
+'''3:''' For events ''A'' and ''B'', <math>P(A\cup B)\geq P(A)</math>.
-'''4:''' For events ''A'' and ''B'', <math>P(A\cup B) = P(A) + P(B) - P(AB)</math>.
+'''4:''' For events ''A'' and ''B'', <math>P(A\cap B)\leq P(A)</math>.
-'''4a:''' For events ''A'' and ''B'', <math>P(A\cup B) \leq P(A) + P(B)</math>.  This is usually referred to as the union bound.
+'''5:''' For events ''A'' and ''B'', <math>P(A\cup B) = P(A) + P(B) - P(AB)</math>.
 == Conditional probabilities ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 04:24, 13 กันยายน 2557
แถว 51:		แถว 51:

	== Bayes' formula ==		== Bayes' formula ==
		+	For any events ''A'' and ''B'', if we know <math>P(B\|A)</math> and <math>P(B\|A^{c})</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 04:23, 13 กันยายน 2557
แถว 29:		แถว 29:

	'''4:''' For events ''A'' and ''B'', <math>P(A\cup B) = P(A) + P(B) - P(AB)</math>.		'''4:''' For events ''A'' and ''B'', <math>P(A\cup B) = P(A) + P(B) - P(AB)</math>.
		+
		+	'''4a:''' For events ''A'' and ''B'', <math>P(A\cup B) \leq P(A) + P(B)</math>. This is usually referred to as the union bound.

	== Conditional probabilities ==		== Conditional probabilities ==