Symplectomorphism - Revision history

en>'Ff'lo: /* Formal definition */ a tad more minor copyediting

2014-06-19T13:47:18Z

Formal definition: a tad more minor copyediting

← Older revision		Revision as of 15:47, 19 June 2014
Line 1:		Line 1:
	The proposition in [[probability theory]] known as the '''law of total expectation''',<ref>Neil A. Weiss, ''A Course in Probability'', Addison–Wesley, 2005, pages 380–383.</ref> the '''law of iterated expectations''', the '''tower rule''', the '''smoothing theorem''', among other names, states that if ''X'' is an integrable [[random variable]] (i.e., a random variable satisfying E( \| ''X'' \| ) < ∞) and ''Y'' is any random variable, not necessarily integrable, on the same [[probability space]], then		My name: Loretta Appleroth<br>My age: 30<br>Country: Denmark<br>Home town: Kobenhavn K <br>Post code: 1105<br>Address: Norrebrovanget 98<br><br>My weblog; [http://funfamilycycling.com/node/35998 cattle dog training]

	:<~~math>\operatorname{E} (X) = \operatorname{E}_Y ( \operatorname{E}_{X \mid Y} ( X \mid Y)),</math~~>

	~~i.e., the [[expected value]] of the conditional expected value of ''X'' given ''Y'' is the same as the expected value of ''X''.~~

	~~The nomenclature used here parallels the phrase ''[[law of total probability]]''. See also [[law of total variance]].~~

	(The [[conditional expected value]] E( ''X'' \| ''Y'' ) is a random variable in its own right, whose value depends on the value of ''Y''. Notice that the conditional expected value of ''X'' given the ''event'' ''Y'' = ''y'' is a function of ''y'' (this is where adherence to the conventional, rigidly case-sensitive [[notation in probability\|notation of probability theory]] becomes important!). If we write E( ''X'' \| ''Y'' = ''y'') = ''g''(''y'') then the random variable E( ''X'' \| ''Y'' ) is just ''g''(''Y'').

	~~One special case states that if <math>A_1, A_2, \ldots, A_n</math> is a partition of the whole outcome space, i.e. these events are mutually exclusive and exhaustive, then~~

	:<~~math~~>~~\operatorname{E} (X) = \sum_{i=1}^{n}{\operatorname{E}(X \mid A_i) \operatorname{P}(A_i)}.</math>~~

	~~==Proof in the discrete case==~~
	:<~~math~~>
	~~\begin{align}~~
	~~\operatorname{E}_Y \left( \operatorname{E}_{X\mid Y} (X \mid Y) \right) &{} = \operatorname{E}_Y \Bigg[ \sum_x x \cdot \operatorname{P}(X=x \mid Y) \Bigg] \\[6pt]~~
	~~&{}=\sum_y \Bigg[ \sum_x x \cdot \operatorname{P}(X=x \mid Y=y) \Bigg] \cdot \operatorname{P}(Y=y) \\[6pt]~~
	~~&{}=\sum_y \sum_x x \cdot \operatorname{P}(X=x \mid Y=y) \cdot \operatorname{P}(Y=y) \\[6pt]~~
	~~&{}=\sum_x x \sum_y \operatorname{P}(X=x \mid Y=y) \cdot \operatorname{P}(Y=y) \\[6pt]~~
	~~&{}=\sum_x x \sum_y \operatorname{P}(X=x, Y=y) \\[6pt]~~
	~~&{}=\sum_x x \cdot \operatorname{P}(X=x) \\[6pt]~~
	~~&{}=\operatorname{E}(X).~~
	~~\end{align}~~
	~~</math>~~

	~~==Notational shortcut==~~
	When using the expectation operator <math>\operatorname{E}</math> , precision would require to always specify the variable the expectation of which is taken. In the case of discrete variables for instance,

	:<~~math~~>~~\operatorname{E}_X (X\cdot Y) = \sum_x x\cdot P(X=x) \cdot Y,</math>~~

	~~is most of the time very different from~~

	:<~~math>\operatorname{E}_Y (X\cdot Y) = \sum_y X \cdot y \cdot P(Y=y),</math>~~

	so that simply writing <math> \operatorname{E}(X\cdot Y) </math> may be confusing. However, adding indices to the expectation operator may lead to cumbersome notations and it is often the case that these indices are omitted. One must then determine from the context or from some convention which variable to take the expectation of.

	Such notational shortcut is very common in the case of iterated expectations where <math>\operatorname{E} \left( \operatorname{E} (X \mid Y) \right) </math> usually stands for <math>\operatorname{E}_Y \left( \operatorname{E}_{X\mid Y} (X \mid Y) \right) </math>. By convention, in the notation without indices, the innermost expectation is the conditional expectation of <math>X</math> given <math>Y</math>, and that the outermost expectation is taken with respect to the conditioning variable <math>Y</math>. This convention is notably used in the rest of this article.

	~~==Iterated expectations with nested conditioning sets==~~
	~~The following formulation of the '''law of iterated expectations''' plays an important role in many economic and finance models~~:

	~~:<math>\operatorname{E} (X \mid I_1) = \operatorname{E} ( \operatorname{E} ( X \mid I_2) \mid I_1),~~<~~/math~~>

	~~where the value of ''I''~~<~~sub~~>1</sub> is determined by that of ''I''<sub>2</sub>. To build intuition, imagine an investor who forecasts a random stock price ''X'' based on the limited information set ''I''<sub>1</sub>. The law of iterated expectations says that the investor can never gain a more precise forecast of ''X'' by conditioning on more specific information (''I''<sub>2</sub>), ''if'' the more specific forecast must itself be forecast with the original information (''I''<sub>1</sub>).

	~~This formulation is often applied in a [~~[time series]] context, where E<sub>''t''</sub> denotes expectations conditional on only the information observed up to and including time period ''t''. In typical models the information set ''t'' + 1 contains all information available through time ''t'', plus additional information revealed at time ''t'' + 1. One can then write:<ref>{{cite book\|title=Recursive macroeconomic theory\|authors=Lars Ljungqvist, Thomas J. Sargent\|isbn=978-0-262-12274-0\|url=http://~~books.google~~.com/~~books?id=Xx-j-tYaPQUC&printsec=frontcover&dq=recursive+macro+theory#v=snippet&q=recursions%20on%20equation&f=false}}</ref>~~

	~~:<math>\operatorname{E}_t(X) = \operatorname{E}_t ( \operatorname{E}_{t+1} ( X )).<~~/~~math>~~

	~~==See also==~~
	* [[Law of total probability]]

	~~==References==~~
	~~{{Reflist}}~~
	*{{cite book \| last=Billingsley \| first=Patrick \| title=Probability and measure \| publisher=John Wiley & Sons, Inc \| location=New York, NY \| year=1995 \| isbn=0-471-00710-2}} (Theorem 34.4)
	*http://sims.princeton.edu/yftp/Bubbles2007/ProbNotes.pdf, especially equations (16) through (18)

	~~{{DEFAULTSORT:Law Of Total Expectation}}~~
	~~[[Category:Algebra of random variables]]~~
	~~[[Category:Theory of probability distributions]]~~
	~~[[Category:Statistical laws]~~]

en>Enyokoyama: /* add a link */

2013-10-29T14:55:37Z

add a link

← Older revision		Revision as of 16:55, 29 October 2013
Line 1:		Line 1:
	~~To regain normal cycling piles achieve this by maintaining certain level~~ of ~~hormones and suppressing~~ the ~~point~~ of other ~~hormones to achieve the result. If a woman wants to have a quick pregnancy~~, it'~~s always handy to have~~ an ~~inexpensive ovulation test kit~~. ~~Getting blood to flow can be difficult when~~ a ~~woman~~ is ~~pregnant and by~~ not ~~moving your legs enough~~, ~~you can get blood clots~~, ~~which can be deadly~~. ~~Another popular way to determine ovulation, although it can be considerably more costly~~, is ~~by utilizing ovulation predictor kits~~. The ~~fertility diet program will show you exactly what they are and create one~~'~~s body in a situation~~ of ~~balance~~. ~~<br><br>These are not very dependable indications~~ of ~~ovulation~~, ~~nonetheless, as a result~~ of ~~they will have many causes~~. ~~It may interferes with~~ the ~~embryo implantation if~~ the ~~uterine tissue extends through entire region~~ of the ~~uterine wall resulting~~ in ~~miscarriage~~. If it is ~~greater than 28 days, you~~'~~ll ovulate later~~. ~~If you think you can't have~~ a ~~baby~~, ~~it is possible that your body will act that way~~. ~~If not every day; even every other day will help because sperm can survive inside you for up to five days in readiness for when that egg decides to appear~~. <br><br>~~d do well to start taking supplements containing iron~~, ~~calcium and folic acid to encourage pregnancy and avoid birth defects~~. ~~In fact~~, ~~according~~ to ~~a 2006 CDC report,~~ the ~~average percentage of ART cycles using fresh non-donor eggs that led to a healthy baby were 40. Acupuncture balances~~ the ~~stage~~ of ~~hormones in your body~~ which is ~~incredibly critical for a woman to turn into a mother~~. ~~You can find lots~~ of ~~folks like you and also you will find many approaches which present fine outcomes. a~~) ~~In men~~, ~~it causes low levels~~ of testosterone resulting of small size ****, ~~cryptorchism leading to infertility because of no sperm during ejaculation.~~ <br><br>~~The very same variables that control~~ the ~~birth of a male also help~~ to ~~encourage the birth of a female, only the positions~~ and ~~conditions are altered. This~~ is often ~~well before~~ the ~~mother even knows she is pregnant~~. ~~Some may be surprised that Dion turned~~ to ~~an unconventional treatment such as acupuncture to benefit her IVF experience~~. ~~Hold off on weaning your little one because women who are breastfeeding are nine times more likely to conceive twins compared to mothers who have stopped breastfeeding~~. ~~As estrogen levels carry on to rise~~, ~~there can be a ten-fold advance~~ in the ~~measure~~ of ~~capacity of your cervical mucus.~~ <br><br>~~In fact~~ the ~~baby continues~~ to ~~gain fat at~~ the ~~rate~~ of ~~half~~ an ~~ounce each~~ and ~~every day. Sadly~~, ~~what normally happens with medical science~~ is that the ~~“whole” person is not included~~. ~~Try to make out~~ the ~~day or~~ the ~~days when you are ovulating each month~~. ~~To have a better chance~~ in ~~conceiving~~ a ~~boy~~, the ~~male partner should consider wearing more loose undergarment~~ and ~~jeans~~. ~~According to research,~~ the ~~ideal~~ time ~~to make love is during the ovulation period and any~~ time ~~for that matter~~.<br><br>~~Here's more information about how to get pregnant on the~~ first ~~try~~ ([http://~~www~~.~~quebec-1759~~.~~info~~/ ~~www~~.~~quebec-1759.info~~]~~) take a look at our web page.~~		The proposition in [[probability theory]] known as the '''law of total expectation''',<ref>Neil A. Weiss, ''A Course in Probability'', Addison–Wesley, 2005, pages 380–383.</ref> the '''law of iterated expectations''', the '''tower rule''', the '''smoothing theorem''', among other names, states that if ''X'' is an integrable [[random variable]] (i.e., a random variable satisfying E( \| ''X'' \| ) < ∞) and ''Y'' is any random variable, not necessarily integrable, on the same [[probability space]], then

			:<math>\operatorname{E} (X) = \operatorname{E}_Y ( \operatorname{E}_{X \mid Y} ( X \mid Y)),</math>

			i.e., the [[expected value]] of the conditional expected value of ''X'' given ''Y'' is the same as the expected value of ''X''.

			The nomenclature used here parallels the phrase ''[[law of total probability]]''. See also [[law of total variance]].

			(The [[conditional expected value]] E( ''X'' \| ''Y'' ) is a random variable in its own right, whose value depends on the value of ''Y''. Notice that the conditional expected value of ''X'' given the ''event'' ''Y'' = ''y'' is a function of ''y'' (this is where adherence to the conventional, rigidly case-sensitive [[notation in probability\|notation of probability theory]] becomes important!). If we write E( ''X'' \| ''Y'' = ''y'') = ''g''(''y'') then the random variable E( ''X'' \| ''Y'' ) is just ''g''(''Y'').

			One special case states that if <math>A_1, A_2, \ldots, A_n</math> is a partition of the whole outcome space, i.e. these events are mutually exclusive and exhaustive, then

			:<math>\operatorname{E} (X) = \sum_{i=1}^{n}{\operatorname{E}(X \mid A_i) \operatorname{P}(A_i)}.</math>

			==Proof in the discrete case==
			:<math>
			\begin{align}
			\operatorname{E}_Y \left( \operatorname{E}_{X\mid Y} (X \mid Y) \right) &{} = \operatorname{E}_Y \Bigg[ \sum_x x \cdot \operatorname{P}(X=x \mid Y) \Bigg] \\[6pt]
			&{}=\sum_y \Bigg[ \sum_x x \cdot \operatorname{P}(X=x \mid Y=y) \Bigg] \cdot \operatorname{P}(Y=y) \\[6pt]
			&{}=\sum_y \sum_x x \cdot \operatorname{P}(X=x \mid Y=y) \cdot \operatorname{P}(Y=y) \\[6pt]
			&{}=\sum_x x \sum_y \operatorname{P}(X=x \mid Y=y) \cdot \operatorname{P}(Y=y) \\[6pt]
			&{}=\sum_x x \sum_y \operatorname{P}(X=x, Y=y) \\[6pt]
			&{}=\sum_x x \cdot \operatorname{P}(X=x) \\[6pt]
			&{}=\operatorname{E}(X).
			\end{align}
			</math>

			==Notational shortcut==
			When using the expectation operator <math>\operatorname{E}</math> , precision would require to always specify the variable the expectation of which is taken. In the case of discrete variables for instance,

			:<math>\operatorname{E}_X (X\cdot Y) = \sum_x x\cdot P(X=x) \cdot Y,</math>

			is most of the time very different from

			:<math>\operatorname{E}_Y (X\cdot Y) = \sum_y X \cdot y \cdot P(Y=y),</math>

			so that simply writing <math> \operatorname{E}(X\cdot Y) </math> may be confusing. However, adding indices to the expectation operator may lead to cumbersome notations and it is often the case that these indices are omitted. One must then determine from the context or from some convention which variable to take the expectation of.

			Such notational shortcut is very common in the case of iterated expectations where <math>\operatorname{E} \left( \operatorname{E} (X \mid Y) \right) </math> usually stands for <math>\operatorname{E}_Y \left( \operatorname{E}_{X\mid Y} (X \mid Y) \right) </math>. By convention, in the notation without indices, the innermost expectation is the conditional expectation of <math>X</math> given <math>Y</math>, and that the outermost expectation is taken with respect to the conditioning variable <math>Y</math>. This convention is notably used in the rest of this article.

			==Iterated expectations with nested conditioning sets==
			The following formulation of the '''law of iterated expectations''' plays an important role in many economic and finance models:

			:<math>\operatorname{E} (X \mid I_1) = \operatorname{E} ( \operatorname{E} ( X \mid I_2) \mid I_1),</math>

			where the value of ''I''<sub>1</sub> is determined by that of ''I''<sub>2</sub>. To build intuition, imagine an investor who forecasts a random stock price ''X'' based on the limited information set ''I''<sub>1</sub>. The law of iterated expectations says that the investor can never gain a more precise forecast of ''X'' by conditioning on more specific information (''I''<sub>2</sub>), ''if'' the more specific forecast must itself be forecast with the original information (''I''<sub>1</sub>).

			This formulation is often applied in a [[time series]] context, where E<sub>''t''</sub> denotes expectations conditional on only the information observed up to and including time period ''t''. In typical models the information set ''t'' + 1 contains all information available through time ''t'', plus additional information revealed at time ''t'' + 1. One can then write:<ref>{{cite book\|title=Recursive macroeconomic theory\|authors=Lars Ljungqvist, Thomas J. Sargent\|isbn=978-0-262-12274-0\|url=http://books.google.com/books?id=Xx-j-tYaPQUC&printsec=frontcover&dq=recursive+macro+theory#v=snippet&q=recursions%20on%20equation&f=false}}</ref>

			:<math>\operatorname{E}_t(X) = \operatorname{E}_t ( \operatorname{E}_{t+1} ( X )).</math>

			==See also==
			* [[Law of total probability]]

			==References==
			{{Reflist}}
			*{{cite book \| last=Billingsley \| first=Patrick \| title=Probability and measure \| publisher=John Wiley & Sons, Inc \| location=New York, NY \| year=1995 \| isbn=0-471-00710-2}} (Theorem 34.4)
			*http://sims.princeton.edu/yftp/Bubbles2007/ProbNotes.pdf, especially equations (16) through (18)

			{{DEFAULTSORT:Law Of Total Expectation}}
			[[Category:Algebra of random variables]]
			[[Category:Theory of probability distributions]]
			[[Category:Statistical laws]]

en>Tosha at 21:24, 17 July 2012

2012-07-17T21:24:39Z

New page

To regain normal cycling piles achieve this by maintaining certain level of hormones and suppressing the point of other hormones to achieve the result. If a woman wants to have a quick pregnancy, it's always handy to have an inexpensive ovulation test kit. Getting blood to flow can be difficult when a woman is pregnant and by not moving your legs enough, you can get blood clots, which can be deadly. Another popular way to determine ovulation, although it can be considerably more costly, is by utilizing ovulation predictor kits. The fertility diet program will show you exactly what they are and create one's body in a situation of balance. These are not very dependable indications of ovulation, nonetheless, as a result of they will have many causes. It may interferes with the embryo implantation if the uterine tissue extends through entire region of the uterine wall resulting in miscarriage. If it is greater than 28 days, you'll ovulate later. If you think you can't have a baby, it is possible that your body will act that way. If not every day; even every other day will help because sperm can survive inside you for up to five days in readiness for when that egg decides to appear. d do well to start taking supplements containing iron, calcium and folic acid to encourage pregnancy and avoid birth defects. In fact, according to a 2006 CDC report, the average percentage of ART cycles using fresh non-donor eggs that led to a healthy baby were 40. Acupuncture balances the stage of hormones in your body which is incredibly critical for a woman to turn into a mother. You can find lots of folks like you and also you will find many approaches which present fine outcomes. a) In men, it causes low levels of testosterone resulting of small size ****, cryptorchism leading to infertility because of no sperm during ejaculation. The very same variables that control the birth of a male also help to encourage the birth of a female, only the positions and conditions are altered. This is often well before the mother even knows she is pregnant. Some may be surprised that Dion turned to an unconventional treatment such as acupuncture to benefit her IVF experience. Hold off on weaning your little one because women who are breastfeeding are nine times more likely to conceive twins compared to mothers who have stopped breastfeeding. As estrogen levels carry on to rise, there can be a ten-fold advance in the measure of capacity of your cervical mucus. In fact the baby continues to gain fat at the rate of half an ounce each and every day. Sadly, what normally happens with medical science is that the “whole” person is not included. Try to make out the day or the days when you are ovulating each month. To have a better chance in conceiving a boy, the male partner should consider wearing more loose undergarment and jeans. According to research, the ideal time to make love is during the ovulation period and any time for that matter. Here's more information about how to get pregnant on the first try ([http://www.quebec-1759.info/ www.quebec-1759.info]) take a look at our web page.