Statistical Inference/Probability Theory - Revision history

Root at 22:27, 3 December 2018

2018-12-03T22:27:06Z

Root: /* Base relationships */

2018-12-03T22:13:38Z

Base relationships

← Older revision		Revision as of 22:13, 3 December 2018
Line 19:		Line 19:
	'''Containment:'''		'''Containment:'''

	<~~amsmath~~>A \subset B \Leftrightarrow x \in A \Rightarrow x \in B</~~amsmath~~>		<math>A \subset B \Leftrightarrow x \in A \Rightarrow x \in B</math>

	'''Equality:'''		'''Equality:'''

	<~~amsmath~~>A = B \Leftrightarrow A \subset B \land B \subset A</~~amsmath~~>		<math>A = B \Leftrightarrow A \subset B \land B \subset A</math>

	=== Elementary operations ===		=== Elementary operations ===

Root: /* Factorial */

2018-08-07T15:40:28Z

Factorial

← Older revision		Revision as of 15:40, 7 August 2018
Line 160:		Line 160:
	=== Factorial ===		=== Factorial ===

	For a positive integer n, n! (read n factorial) is the product of all of the positive integers less than or equal to n [Definition 1.2.16]. That is, <amsmath>nl = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1</amsmath>. Furthermore, we define 0! = 1.		For a positive integer n, n! (read n factorial) is the product of all of the positive integers less than or equal to n [Definition 1.2.16]. That is, <amsmath>n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1</amsmath>. Furthermore, we define 0! = 1.

	== Enumerating Outcomes ==		== Enumerating Outcomes ==

Root: /* Counting */

2018-08-05T09:37:51Z

Counting

← Older revision		Revision as of 09:37, 5 August 2018
Line 146:		Line 146:
	== Counting ==		== Counting ==

	Most often, methods of counting are used in order to construct probability assignments on ~~fmite~~ sample spaces, although they can be used to answer other questions also.		Most often, methods of counting are used in order to construct probability assignments on finite sample spaces, although they can be used to answer other questions also.

	Counting problems, in general, sound complicated, and often we must do our counting subject to many restrictions. The way to solve such problems is to break them down into a series of simple tasks that are easy to count, and employ known rules of combining tasks. The following theorem is a first step in such a process and is sometimes known as the Fundamental Theorem of Counting.		Counting problems, in general, sound complicated, and often we must do our counting subject to many restrictions. The way to solve such problems is to break them down into a series of simple tasks that are easy to count, and employ known rules of combining tasks. The following theorem is a first step in such a process and is sometimes known as the Fundamental Theorem of Counting.

Root: /* Counting */

2018-08-04T11:46:17Z

Counting

← Older revision		Revision as of 11:46, 4 August 2018
Line 153:		Line 153:

	If a job consists of <amsmath>k</amsmath> separate tasks, the <amsmath>i</amsmath>-th of which can be done in <amsmath>n_i</amsmath> ways, <amsmath>i = 1, \dots, k</amsmath>, then the entire job can be done in <amsmath>n_1 \times n_2 \times \dots \times n_k</amsmath> ways.		If a job consists of <amsmath>k</amsmath> separate tasks, the <amsmath>i</amsmath>-th of which can be done in <amsmath>n_i</amsmath> ways, <amsmath>i = 1, \dots, k</amsmath>, then the entire job can be done in <amsmath>n_1 \times n_2 \times \dots \times n_k</amsmath> ways.

			=== Possible Methods of Counting ===

			Counting can be made with without replacement. Also, the order in which the task is performed may matter. Four types counting scenarios are thus possible: ordered or unordered, both of which may be done with or without replacement.

			=== Factorial ===

			For a positive integer n, n! (read n factorial) is the product of all of the positive integers less than or equal to n [Definition 1.2.16]. That is, <amsmath>nl = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1</amsmath>. Furthermore, we define 0! = 1.

	== Enumerating Outcomes ==		== Enumerating Outcomes ==

Root: /* Counting */

2018-08-04T11:01:43Z

Counting

← Older revision		Revision as of 11:01, 4 August 2018
Line 147:		Line 147:

	Most often, methods of counting are used in order to construct probability assignments on fmite sample spaces, although they can be used to answer other questions also.		Most often, methods of counting are used in order to construct probability assignments on fmite sample spaces, although they can be used to answer other questions also.

			Counting problems, in general, sound complicated, and often we must do our counting subject to many restrictions. The way to solve such problems is to break them down into a series of simple tasks that are easy to count, and employ known rules of combining tasks. The following theorem is a first step in such a process and is sometimes known as the Fundamental Theorem of Counting.

			=== Fundamental Theorem of Counting ===

			If a job consists of <amsmath>k</amsmath> separate tasks, the <amsmath>i</amsmath>-th of which can be done in <amsmath>n_i</amsmath> ways, <amsmath>i = 1, \dots, k</amsmath>, then the entire job can be done in <amsmath>n_1 \times n_2 \times \dots \times n_k</amsmath> ways.

	== Enumerating Outcomes ==		== Enumerating Outcomes ==

Root: /* Counting */

2018-08-04T09:31:36Z

Counting

← Older revision		Revision as of 09:31, 4 August 2018
Line 147:		Line 147:

	Most often, methods of counting are used in order to construct probability assignments on fmite sample spaces, although they can be used to answer other questions also.		Most often, methods of counting are used in order to construct probability assignments on fmite sample spaces, although they can be used to answer other questions also.

			== Enumerating Outcomes ==

	= Conditional Probability and Independence =		= Conditional Probability and Independence =

Root: /* Properties of the probability function applied to collections of sets */

2018-08-01T18:22:21Z

Properties of the probability function applied to collections of sets

← Older revision		Revision as of 18:22, 1 August 2018
Line 138:		Line 138:
	If P is a probability function, then [Theorem 1.2.11]		If P is a probability function, then [Theorem 1.2.11]
	* (a) <amsmath>P(A) = \Sigma^\infty_{i=1} P(A \cap C_i)</amsmath>, for any partition <amsmath>C_1, C_2, \dots</amsmath>;		* (a) <amsmath>P(A) = \Sigma^\infty_{i=1} P(A \cap C_i)</amsmath>, for any partition <amsmath>C_1, C_2, \dots</amsmath>;
	* (b) <amsmath>P(\cup^\infty_{i=1} A_i) \le \Sigma^\infty_{i=1} P(A_i)</amsmath>, for any sets <amsmath>A_1, A_2, \dots</amsmath> (~~Booleí~~'s Inequality)		* (b) <amsmath>P(\cup^\infty_{i=1} A_i) \le \Sigma^\infty_{i=1} P(A_i)</amsmath>, for any sets <amsmath>A_1, A_2, \dots</amsmath> (Boole's Inequality)

	There is a similarity between Boole's Inequality and Bonferroni's Inequality. In fact, they are essentially the same thing. We could have used Boole's Inequality to derive that <amsmath>P(A \cap B) \ge P(A) + P(B) - 1</amsmath> (the special case of Bonferroni's Inequality, presented above). If we apply Boole*s Inequality to <amsmath>\overline{A}</amsmath>, we have <amsmath>P(\cup^n_{i=1} \overline{A_i}) \le \Sigma^n_{i=1} P(\overline{A_i})</amsmath> and using the facts that <amsmath>\cup\overline{A_i} = \overline{\cap A_i}</amsmath> and <amsmath>P(\overline{A_i}) = 1 - P(A_i)</amsmath>, we obtain <amsmath>1 - P(\cap^n_{i=1} A_i) \le n - \Sigma^n_{i=1} P(A_i)</amsmath>		There is a similarity between Boole's Inequality and Bonferroni's Inequality. In fact, they are essentially the same thing. We could have used Boole's Inequality to derive that <amsmath>P(A \cap B) \ge P(A) + P(B) - 1</amsmath> (the special case of Bonferroni's Inequality, presented above). If we apply Boole*s Inequality to <amsmath>\overline{A}</amsmath>, we have <amsmath>P(\cup^n_{i=1} \overline{A_i}) \le \Sigma^n_{i=1} P(\overline{A_i})</amsmath> and using the facts that <amsmath>\cup\overline{A_i} = \overline{\cap A_i}</amsmath> and <amsmath>P(\overline{A_i}) = 1 - P(A_i)</amsmath>, we obtain <amsmath>1 - P(\cap^n_{i=1} A_i) \le n - \Sigma^n_{i=1} P(A_i)</amsmath>

Root: /* The Calculus of Probabilities */

2018-08-01T18:21:04Z

The Calculus of Probabilities

← Older revision		Revision as of 18:21, 1 August 2018
Line 133:		Line 133:

	This inequality is a special case of what is known as Bonferroni's Inequality. Bonferroni's Inequality allows us to bound the probability of a simultaneous event (the intersectíon) in terms of the probabilities of the individual events.		This inequality is a special case of what is known as Bonferroni's Inequality. Bonferroni's Inequality allows us to bound the probability of a simultaneous event (the intersectíon) in terms of the probabilities of the individual events.

			=== Properties of the probability function applied to collections of sets ===

			If P is a probability function, then [Theorem 1.2.11]
			* (a) <amsmath>P(A) = \Sigma^\infty_{i=1} P(A \cap C_i)</amsmath>, for any partition <amsmath>C_1, C_2, \dots</amsmath>;
			* (b) <amsmath>P(\cup^\infty_{i=1} A_i) \le \Sigma^\infty_{i=1} P(A_i)</amsmath>, for any sets <amsmath>A_1, A_2, \dots</amsmath> (Booleí's Inequality)

			There is a similarity between Boole's Inequality and Bonferroni's Inequality. In fact, they are essentially the same thing. We could have used Boole's Inequality to derive that <amsmath>P(A \cap B) \ge P(A) + P(B) - 1</amsmath> (the special case of Bonferroni's Inequality, presented above). If we apply Boole*s Inequality to <amsmath>\overline{A}</amsmath>, we have <amsmath>P(\cup^n_{i=1} \overline{A_i}) \le \Sigma^n_{i=1} P(\overline{A_i})</amsmath> and using the facts that <amsmath>\cup\overline{A_i} = \overline{\cap A_i}</amsmath> and <amsmath>P(\overline{A_i}) = 1 - P(A_i)</amsmath>, we obtain <amsmath>1 - P(\cap^n_{i=1} A_i) \le n - \Sigma^n_{i=1} P(A_i)</amsmath>

			This becomes, on rearranging terms, <amsmath>P(\cap^n_{i=1} A_i) \ge \Sigma^n_{i=1} P(A_i) - (n -1)</amsmath>, which is a more general version of the Bonferroni Inequality presented before.

			== Counting ==

			Most often, methods of counting are used in order to construct probability assignments on fmite sample spaces, although they can be used to answer other questions also.

	= Conditional Probability and Independence =		= Conditional Probability and Independence =

Root: /* Bonferroní's Inequality */

2018-08-01T17:50:13Z

Bonferroní's Inequality

← Older revision		Revision as of 17:50, 1 August 2018
Line 128:		Line 128:
	* (c) If <amsmath>A \subset B</amsmath>, then <amsmath>P(A) \le P(B)</amsmath>.		* (c) If <amsmath>A \subset B</amsmath>, then <amsmath>P(A) \le P(B)</amsmath>.

	=== ~~Bonferroní~~'s Inequality ===		=== Bonferroni's Inequality ===

	Formula (b) of Theorem 1.2.9 gives a useful inequality for the probability of an intersection. Since <amsmath>P(A \cup B) \le 1</amsmath>, we have, after some rearranging, <amsmath>P(A \cap B) \ge P(A) + P(B) - 1</amsmath>.		Formula (b) of Theorem 1.2.9 gives a useful inequality for the probability of an intersection. Since <amsmath>P(A \cup B) \le 1</amsmath>, we have, after some rearranging, <amsmath>P(A \cap B) \ge P(A) + P(B) - 1</amsmath>.

	This inequality is a special case of what is known as ~~Bonferroní~~'s Inequality. ~~Bonferroní~~'s Inequality allows us to bound the probability of a simultaneous event (the intersectíon) in terms of the probabilities of the individual events.		This inequality is a special case of what is known as Bonferroni's Inequality. Bonferroni's Inequality allows us to bound the probability of a simultaneous event (the intersectíon) in terms of the probabilities of the individual events.

	= Conditional Probability and Independence =		= Conditional Probability and Independence =