Variance - Revision history

IssaRice: /* Questions/things to explain */

2019-07-31T00:36:54Z

Questions/things to explain

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	* vector space interpretation [https://math.stackexchange.com/a/3071375/35525] see also the beginning of [https://web.math.princeton.edu/~nelson/books/rept.pdf]		* vector space interpretation [https://math.stackexchange.com/a/3071375/35525] see also the beginning of [https://web.math.princeton.edu/~nelson/books/rept.pdf]
			* is there some list of axioms that can completely specify the variance? (similar to how the Riemann integral, plane geometry, various algebraic structures, etc., can all be defined axiomatically)
			* because of the square in the definition of variance, we have also squared the units. To get back the units we started with, one possibility is to take the square root of the variance. This leads to the standard deviation.
			* why not divide by the mean (the same data measured in another units system has different variance)?

	[[Category:Probability]]		[[Category:Probability]]

IssaRice: /* Motivation */

2019-07-31T00:33:25Z

Motivation

← Older revision		Revision as of 00:33, 31 July 2019
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	# We want to measure the spread of the data. For each data point, we can subtract the mean from it to see how "deviant" it is. So one a priori reasonable approach is to calculate these deviations and find the average deviation. If we do this, however, we get <math>\mathbf E[X - \mathbf E X] = \mathbf E X - \mathbf E X = 0</math> which is useless.		# We want to measure the spread of the data. For each data point, we can subtract the mean from it to see how "deviant" it is. So one a priori reasonable approach is to calculate these deviations and find the average deviation. If we do this, however, we get <math>\mathbf E[X - \mathbf E X] = \mathbf E X - \mathbf E X = 0</math> which is useless.
	# We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get <math>\mathbf E \|X - \mathbf E X\|</math>. But the absolute value is not smooth enough for us to be able to do all the things we would like (e.g. differentiation). However, note that this measure of the spread is also used.		# We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get <math>\mathbf E \|X - \mathbf E X\|</math>. But the absolute value is not smooth enough for us to be able to do all the things we would like (e.g. differentiation). However, note that this measure of the spread is also used (what is it called?).
	# Finally, building off the previous idea, we decide to square things, and we get <math>\mathbf E [(X - \mathbf E X)^2]</math>, which is the definition of variance.		# Finally, building off the previous idea, we decide to square things, and we get <math>\mathbf E [(X - \mathbf E X)^2]</math>, which is the definition of variance.

IssaRice: /* Motivation */

2019-01-16T06:14:30Z

Motivation

← Older revision		Revision as of 06:14, 16 January 2019
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	# We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get <math>\mathbf E \|X - \mathbf E X\|</math>. But the absolute value is not smooth enough for us to be able to do all the things we would like (e.g. differentiation). However, note that this measure of the spread is also used.		# We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get <math>\mathbf E \|X - \mathbf E X\|</math>. But the absolute value is not smooth enough for us to be able to do all the things we would like (e.g. differentiation). However, note that this measure of the spread is also used.
	# Finally, building off the previous idea, we decide to square things, and we get <math>\mathbf E [(X - \mathbf E X)^2]</math>, which is the definition of variance.		# Finally, building off the previous idea, we decide to square things, and we get <math>\mathbf E [(X - \mathbf E X)^2]</math>, which is the definition of variance.

			If we can intuitively understand [[covariance]] (I'm still working on this understanding), then we can get the variance as <math>\operatorname{Cov}(X,X)</math>.

	==Questions/things to explain==		==Questions/things to explain==

IssaRice: /* Motivation */

2019-01-16T05:44:55Z

Motivation

← Older revision		Revision as of 05:44, 16 January 2019
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	# We want to measure the spread of the data. For each data point, we can subtract the mean from it to see how "deviant" it is. So one a priori reasonable approach is to calculate these deviations and find the average deviation. If we do this, however, we get <math>\mathbf E[X - \mathbf E X] = \mathbf E X - \mathbf E X = 0</math> which is useless.		# We want to measure the spread of the data. For each data point, we can subtract the mean from it to see how "deviant" it is. So one a priori reasonable approach is to calculate these deviations and find the average deviation. If we do this, however, we get <math>\mathbf E[X - \mathbf E X] = \mathbf E X - \mathbf E X = 0</math> which is useless.
	# We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get <math>\mathbf E \|X - \mathbf E X\|</math>. But the absolute value is not smooth enough for us to be able to do all the things we would like. However, note that this measure of the spread is also used.		# We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get <math>\mathbf E \|X - \mathbf E X\|</math>. But the absolute value is not smooth enough for us to be able to do all the things we would like (e.g. differentiation). However, note that this measure of the spread is also used.
	# Finally, building off the previous idea, we decide to square things, and we get <math>\mathbf E [(X - \mathbf E X)^2]</math>, which is the definition of variance.		# Finally, building off the previous idea, we decide to square things, and we get <math>\mathbf E [(X - \mathbf E X)^2]</math>, which is the definition of variance.

IssaRice at 05:43, 16 January 2019

2019-01-16T05:43:34Z

← Older revision		Revision as of 05:43, 16 January 2019
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	Since the square root of the variance is the standard deviation, if we have a simple notation for the standard deviation, such as <math>\sigma</math>, then we can denote the variance as <math>\sigma^2</math>.		Since the square root of the variance is the standard deviation, if we have a simple notation for the standard deviation, such as <math>\sigma</math>, then we can denote the variance as <math>\sigma^2</math>.

			==Motivation==

			In several books I have seen the following three-step motivation for the variance. I'm not sure I'm convinced this is all that can be said to motivate the variance, but it seems to be a start.

			# We want to measure the spread of the data. For each data point, we can subtract the mean from it to see how "deviant" it is. So one a priori reasonable approach is to calculate these deviations and find the average deviation. If we do this, however, we get <math>\mathbf E[X - \mathbf E X] = \mathbf E X - \mathbf E X = 0</math> which is useless.
			# We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get <math>\mathbf E \|X - \mathbf E X\|</math>. But the absolute value is not smooth enough for us to be able to do all the things we would like. However, note that this measure of the spread is also used.
			# Finally, building off the previous idea, we decide to square things, and we get <math>\mathbf E [(X - \mathbf E X)^2]</math>, which is the definition of variance.

	==Questions/things to explain==		==Questions/things to explain==

IssaRice at 03:52, 16 January 2019

2019-01-16T03:52:24Z

← Older revision		Revision as of 03:52, 16 January 2019
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	Since the square root of the variance is the standard deviation, if we have a simple notation for the standard deviation, such as <math>\sigma</math>, then we can denote the variance as <math>\sigma^2</math>.		Since the square root of the variance is the standard deviation, if we have a simple notation for the standard deviation, such as <math>\sigma</math>, then we can denote the variance as <math>\sigma^2</math>.

			==Questions/things to explain==

			* vector space interpretation [https://math.stackexchange.com/a/3071375/35525] see also the beginning of [https://web.math.princeton.edu/~nelson/books/rept.pdf]

	[[Category:Probability]]		[[Category:Probability]]

IssaRice: Created page with "The variance of a random variable $X$ is defined as $\mathrm{Var}(X) := \mathbf E[(X - \mathbf EX)^2]$ , where $\mathbf EX$ is the expectatio..."

2019-01-16T03:48:36Z

Created page with "The variance of a random variable <math>X</math> is defined as <math>\mathrm{Var}(X) := \mathbf E[(X - \mathbf EX)^2]</math>, where <math>\mathbf EX</math> is the expectatio..."

New page

The variance of a random variable <math>X</math> is defined as <math>\mathrm{Var}(X) := \mathbf E[(X - \mathbf EX)^2]</math>, where <math>\mathbf EX</math> is the [[expectation]] of <math>X</math>.

==Notation==

Since the square root of the variance is the standard deviation, if we have a simple notation for the standard deviation, such as <math>\sigma</math>, then we can denote the variance as <math>\sigma^2</math>.

[[Category:Probability]]

Variance - Revision history

IssaRice: /* Questions/things to explain */

IssaRice: /* Motivation */

IssaRice: /* Motivation */

IssaRice: /* Motivation */

IssaRice at 05:43, 16 January 2019

IssaRice at 03:52, 16 January 2019

IssaRice: Created page with "The variance of a random variable X is defined as \mathrm{Var}(X) := \mathbf E[(X - \mathbf EX)^2], where \mathbf EX is the expectatio..."

IssaRice: Created page with "The variance of a random variable $X$ is defined as $\mathrm{Var}(X) := \mathbf E[(X - \mathbf EX)^2]$ , where $\mathbf EX$ is the expectatio..."