Uncertainty - Revision history

Lfox: /* Functoriality */

2024-10-23T01:30:33Z

Functoriality

← Older revision		Revision as of 01:30, 23 October 2024
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	The above construction generalizes vastly.		The above construction generalizes vastly.

	First, let <math>X,Y</math> be any measurable spaces, and let <math>f : X\rightarrow Y</math> be a measurable function. Let <math>\Gamma_X, \Gamma_Y </math> be triangulations of <math>X</math> and <math>Y</math>. Then <math>f</math> induces a map <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_Y)</math>.		First, let <math>X,Y</math> be any measurable spaces, and let <math>f : X\rightarrow Y</math> be a measurable function. Let <math>\Gamma_X, \Gamma_Y </math> be triangulations of <math>X</math> and <math>Y</math>. Then <math>f</math> induces a map <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_Y)</math>. [TODO this isn't exactly right. I need a pre-existing "volume" measure on <math>X</math> in order to get the map <math>\Gamma \rightarrow \text{Prob}(\Gamma)</math>. ]

	Now, if we haven't measured a quantity recently, then it might be right to think about it as a probability distribution, rather than as having a specific value. (It ''does'' have a specific value, but we don't know what it is.) In such a case as this, we can still talk about what a function <math>f : X \rightarrow Y</math> will do to the quantity. Indeed, if the quantity is in a specific cell of <math>\Gamma_X</math>, then we know what <math>f</math> will do to it (in the generalized sense of probability distributions). And the quantity must be in ''some'' specific cell of <math>\Gamma_X</math>. So we should expect <math>f</math> to induce a map <math>\text{Prob}(\Gamma_X) \rightarrow \text{Prob}(\Gamma_Y)</math>, which agrees with <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_Y)</math> when restricted via <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_X)</math>.		Now, if we haven't measured a quantity recently, then it might be right to think about it as a probability distribution, rather than as having a specific value. (It ''does'' have a specific value, but we don't know what it is.) In such a case as this, we can still talk about what a function <math>f : X \rightarrow Y</math> will do to the quantity. Indeed, if the quantity is in a specific cell of <math>\Gamma_X</math>, then we know what <math>f</math> will do to it (in the generalized sense of probability distributions). And the quantity must be in ''some'' specific cell of <math>\Gamma_X</math>. So we should expect <math>f</math> to induce a map <math>\text{Prob}(\Gamma_X) \rightarrow \text{Prob}(\Gamma_Y)</math>, which agrees with <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_Y)</math> when restricted via <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_X)</math>.

			There is a map <math>\alpha : \text{Prob}(\Gamma_X) \rightarrow \text{Prob}(X) </math>, defined as follows:<math display="block">\alpha(p) := \left(A \mapsto \sum_{\Delta \in \Gamma_X} p(\Delta) \frac{\text{Vol}(A \cap \Delta)}{\text{Vol}(\Delta) }\right).</math>This <math>\alpha</math> satisfies the property that <math>\alpha(p)\|_{\Gamma_X} = p</math>, but the composition with restriction in the other direction is most definitely ''not'' the identity. [TODO I expect that the composition in the other direction is "homotopic" to the identity, because it only differs from the original in a local way.]

			So we get a map <math>\rho_Y \circ f_* \circ \alpha_X : \text{Prob}(\Gamma_X) \rightarrow \text{Prob}(\Gamma_Y)</math> where <math>\rho_Y : \text{Prob}(Y) \rightarrow \text{Prob}(\Gamma_Y)</math> is the restriction, whenever we have an <math>f : X \rightarrow Y</math>. However, this construction is not functorial, because <math>\alpha \circ \rho \neq \text{Id}.</math> Another reason it can't be functorial is that "triangulations" are not part of the initial data. Yet another reason it can't be functorial is that the identity map on <math>X</math> can actually give interesting maps between different triangulations of <math>X</math>.

	== Renormalization ==		== Renormalization ==

Lfox at 00:12, 23 October 2024

2024-10-23T00:12:02Z

← Older revision		Revision as of 00:12, 23 October 2024
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	For an interval <math>I_n</math> of <math>\Gamma</math>, let <math>\mu_n</math> be the probability measure defined by <math display="inline">\mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } </math>. Let <math>f : \mathbb{R} \rightarrow \mathbb{R} </math> be any measurable function. Then <math>f </math> induces a probability measure <math>f_* \mu_n </math> on <math>\mathbb{R} </math>, and hence a probability measure <math>f_* \mu_n \|_\Gamma </math> on <math>\Gamma </math>. In this language, <math>\widetilde{f} </math> is simply the function sending <math display="block">\~~begin~~{~~align~~}		For an interval <math>I_n</math> of <math>\Gamma</math>, let <math>\mu_n</math> be the probability measure defined by <math display="inline">\mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } </math>. Let <math>f : \mathbb{R} \rightarrow \mathbb{R} </math> be any measurable function. Then <math>f </math> induces a probability measure <math>f_* \mu_n </math> on <math>\mathbb{R} </math>, and hence a probability measure <math>f_* \mu_n \|_\Gamma </math> on <math>\Gamma </math>. In this language, <math>\widetilde{f} : \Gamma \rightarrow \text{Prob}(\Gamma) </math> is simply the function sending <math display="block">I_n \mapsto (f_* \mu_n) \|_\Gamma . </math>One can easily check that the two formulas for <math>\widetilde{f}</math> agree.
	\~~widetilde~~{f} : \~~Gamma &~~ \rightarrow \text{Prob}(\~~Gamma~~) \\
	~~I_n &~~ \~~mapsto~~ (f_* \~~mu_n~~) \|_\~~Gamma~~		== Functoriality ==
	\~~end~~{~~align~~}. </math>~~One can easily check that the formulas agree~~.		The above construction generalizes vastly.

			First, let <math>X,Y</math> be any measurable spaces, and let <math>f : X\rightarrow Y</math> be a measurable function. Let <math>\Gamma_X, \Gamma_Y </math> be triangulations of <math>X</math> and <math>Y</math>. Then <math>f</math> induces a map <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_Y)</math>.

			Now, if we haven't measured a quantity recently, then it might be right to think about it as a probability distribution, rather than as having a specific value. (It ''does'' have a specific value, but we don't know what it is.) In such a case as this, we can still talk about what a function <math>f : X \rightarrow Y</math> will do to the quantity. Indeed, if the quantity is in a specific cell of <math>\Gamma_X</math>, then we know what <math>f</math> will do to it (in the generalized sense of probability distributions). And the quantity must be in ''some'' specific cell of <math>\Gamma_X</math>. So we should expect <math>f</math> to induce a map <math>\text{Prob}(\Gamma_X) \rightarrow \text{Prob}(\Gamma_Y)</math>, which agrees with <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_Y)</math> when restricted via <math>\Gamma_X \rightarrow \text{Prob}(\Gamma_X)</math>.

	== Renormalization ==		== Renormalization ==

Lfox: /* Measure-theoretic interpretation */

2024-10-22T23:25:49Z

Measure-theoretic interpretation

← Older revision		Revision as of 23:25, 22 October 2024
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	\widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\		\widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\
	I_n & \mapsto (f_* \mu_n) \|_\Gamma		I_n & \mapsto (f_* \mu_n) \|_\Gamma
	\end{align} </math>		\end{align}. </math>One can easily check that the formulas agree.

	== Renormalization ==		== Renormalization ==

Lfox: /* General formulation */

2024-10-22T23:24:39Z

General formulation

← Older revision		Revision as of 23:24, 22 October 2024
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	=== Measure-theoretic interpretation ===		=== Measure-theoretic interpretation ===
	For my purposes in this section, by a measure on <math>\mathbb{R}</math> I mean one for which all points have measure 0. <math>\Gamma</math> is a countable cover of <math>\mathbb{R}</math> by measurable sets. Any measure <math>\mu</math> on <math>\mathbb{R}</math> restricts to a measure <math>\mu \|_\Gamma</math> on <math>\Gamma</math>, where <math display="inline">\mu\|_{\Gamma}(S) := \mu \left( \bigcup_{s\in S} s \right) = \sum_{s\in S} \mu(s). </math> In particular, if <math>\mu</math> is a probability measure then so is <math>\mu \|_\Gamma</math>.		For my purposes in this section, by a measure on <math>\mathbb{R}</math> I mean one for which all points have measure 0. <math>\Gamma</math> is a countable cover of <math>\mathbb{R}</math> by measurable sets. Any measure <math>\mu</math> on <math>\mathbb{R}</math> restricts to a measure <math>\mu \|_\Gamma</math> on <math>\Gamma</math>, where <math display="inline">\mu\|_{\Gamma}(S) := \mu \left( \bigcup_{s\in S} s \right) = \sum_{s\in S} \mu(s). </math> In particular, if <math>\mu</math> is a probability measure then so is <math>\mu \|_\Gamma</math>.



	For an interval <math>I_n</math> of <math>\Gamma</math>, let <math>\mu_n</math> be the probability measure defined by <math display="inline">\mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } </math>. Let <math>f : \mathbb{R} \rightarrow \mathbb{R} </math> be any measurable function. Then <math>f </math> induces a probability measure <math>f_* \mu_n </math> on <math>\mathbb{R} </math>, and hence a probability measure <math>f_* \mu_n \|_\Gamma </math> on <math>\Gamma </math>. In this language, <math>\widetilde{f} </math> is simply the function sending <math display="block">\begin{align}		For an interval <math>I_n</math> of <math>\Gamma</math>, let <math>\mu_n</math> be the probability measure defined by <math display="inline">\mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } </math>. Let <math>f : \mathbb{R} \rightarrow \mathbb{R} </math> be any measurable function. Then <math>f </math> induces a probability measure <math>f_* \mu_n </math> on <math>\mathbb{R} </math>, and hence a probability measure <math>f_* \mu_n \|_\Gamma </math> on <math>\Gamma </math>. In this language, <math>\widetilde{f} </math> is simply the function sending <math display="block">\begin{align}
	\widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\		\widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\
	n & \mapsto (f_* \mu_n) \|_\Gamma		I_n & \mapsto (f_* \mu_n) \|_\Gamma
	\end{align} </math>		\end{align} </math>

Lfox: /* General formulation */

2024-10-22T23:23:33Z

General formulation

← Older revision		Revision as of 23:23, 22 October 2024
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	'''Definition'''. A ''probability distribution'' on a set <math>S</math> is a function <math>p: S \rightarrow [0,1]</math>, such that <math>p_s = 0 </math> for all but finitely many <math>s \in S</math>, and <math>\sum_{s\in S} p_s = 1</math>. Let <math>\text{Prob}(S)</math> denote the set of probability distributions on <math>S</math>.		'''Definition'''. A ''probability distribution'' on a set <math>S</math> is a function <math>p: S \rightarrow [0,1]</math>, such that <math>p_s = 0 </math> for all but finitely many <math>s \in S</math>, and <math>\sum_{s\in S} p_s = 1</math>. Let <math>\text{Prob}(S)</math> denote the set of probability distributions on <math>S</math>.

	Since <math>f</math> is continuous, it sends intervals to intervals. If <math>I = [a,b]</math> is any interval, define its volume as <math>\text{Vol}(I):= b - a</math>. Then <math>f</math> induces a function <math>\widetilde{f} : \Gamma \rightarrow \text{Prob}(\Gamma) </math>, where <math display="block">\widetilde{f}(n)_m := \frac{\text{Vol}(f([a_n, b_n]) \cap [a_m,b_m] ) }{\text{Vol}(f([a_n,b_n]) )}</math>

			Since <math>f</math> is continuous, it sends intervals to intervals. If <math>I = [a,b]</math> is any interval, define its volume as <math>\text{Vol}(I):= b - a</math>. Then <math>f</math> induces a function <math>\widetilde{f} : \Gamma \rightarrow \text{Prob}(\Gamma) </math>, where <math display="block">\widetilde{f}(n)_m := \frac{\text{Vol}([a_n, b_n] \cap f^{-1}([a_m,b_m]) ) }{\text{Vol}([a_n,b_n])} </math>

	The interpretation of <math>\widetilde{f}</math> is completely clear: If the function <math>f</math> is applied to a quantity that was measured as being in the grid interval <math>[a_n, b_n]</math>, then the probability that the resulting quantity will be measured as being in the grid interval <math>[a_m, b_m]</math> is <math>\widetilde{f}(n)_m</math>.		The interpretation of <math>\widetilde{f}</math> is completely clear: If the function <math>f</math> is applied to a quantity that was measured as being in the grid interval <math>[a_n, b_n]</math>, then the probability that the resulting quantity will be measured as being in the grid interval <math>[a_m, b_m]</math> is <math>\widetilde{f}(n)_m</math>.
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	We can say that "a measurement of <math>f</math> relative to the triangulation <math>\Gamma</math>" is a field of random variables, one at each triangle, where the random variable at triangle <math>n</math> is drawn from the probability distribution <math>\widetilde{f}(n)</math>.		We can say that "a measurement of <math>f</math> relative to the triangulation <math>\Gamma</math>" is a field of random variables, one at each triangle, where the random variable at triangle <math>n</math> is drawn from the probability distribution <math>\widetilde{f}(n)</math>.

	~~[TODO] This~~ is ~~probably the same concept as pushing forward~~ a ~~uniform~~ measure ~~localized at~~ <math>~~[a_n~~, ~~b_n]~~</math> by <math>f</math>, ~~but I haven't worked out~~ the ~~details.~~		=== Measure-theoretic interpretation ===
			For my purposes in this section, by a measure on <math>\mathbb{R}</math> I mean one for which all points have measure 0. <math>\Gamma</math> is a countable cover of <math>\mathbb{R}</math> by measurable sets. Any measure <math>\mu</math> on <math>\mathbb{R}</math> restricts to a measure <math>\mu \|_\Gamma</math> on <math>\Gamma</math>, where <math display="inline">\mu\|_{\Gamma}(S) := \mu \left( \bigcup_{s\in S} s \right) = \sum_{s\in S} \mu(s). </math> In particular, if <math>\mu</math> is a probability measure then so is <math>\mu \|_\Gamma</math>.

			For an interval <math>I_n</math> of <math>\Gamma</math>, let <math>\mu_n</math> be the probability measure defined by <math display="inline">\mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } </math>. Let <math>f : \mathbb{R} \rightarrow \mathbb{R} </math> be any measurable function. Then <math>f </math> induces a probability measure <math>f_* \mu_n </math> on <math>\mathbb{R} </math>, and hence a probability measure <math>f_* \mu_n \|_\Gamma </math> on <math>\Gamma </math>. In this language, <math>\widetilde{f} </math> is simply the function sending <math display="block">\begin{align}
			\widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\
			n & \mapsto (f_* \mu_n) \|_\Gamma
			\end{align} </math>

	== Renormalization ==		== Renormalization ==

Lfox at 22:27, 22 October 2024

2024-10-22T22:27:10Z

← Older revision		Revision as of 22:27, 22 October 2024
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	The interpretation of <math>\widetilde{f}</math> is completely clear: If the function <math>f</math> is applied to a quantity that was measured as being in the grid interval <math>[a_n, b_n]</math>, then the probability that the resulting quantity will be measured as being in the grid interval <math>[a_m, b_m]</math> is <math>\widetilde{f}(n)_m</math>.		The interpretation of <math>\widetilde{f}</math> is completely clear: If the function <math>f</math> is applied to a quantity that was measured as being in the grid interval <math>[a_n, b_n]</math>, then the probability that the resulting quantity will be measured as being in the grid interval <math>[a_m, b_m]</math> is <math>\widetilde{f}(n)_m</math>.

			We can say that "a measurement of <math>f</math> relative to the triangulation <math>\Gamma</math>" is a field of random variables, one at each triangle, where the random variable at triangle <math>n</math> is drawn from the probability distribution <math>\widetilde{f}(n)</math>.

			[TODO] This is probably the same concept as pushing forward a uniform measure localized at <math>[a_n, b_n]</math> by <math>f</math>, but I haven't worked out the details.

			== Renormalization ==
			Let <math>f : \mathbb{R} \rightarrow \mathbb{R}</math> be some continuum function. There is a relationship between <math>f</math> as measured on a fine triangulation, and <math>f</math> as measured on a coarse triangulation.

			=== Motivating example ===
			Let's suppose that we are measuring lengths with two meter sticks, one which goes down to the nearest 2 millimeters, and one which goes down to the nearest 1 millimeter. That is, the ticks on the first stick appear at 0mm, 2mm, 4mm, 6mm, etc., and the ticks on the second stick appear at 0mm, 1mm, 2mm, 3mm, etc. The former stick defines a triangulation which we will call <math>\Gamma</math>, and the latter defines a triangulation which we will call <math>\Gamma'</math>.

			There is a (discontinuous!!) function <math>i: \Gamma' \rightarrow \Gamma</math>, sending <math>[0,1] \mapsto [0,2]</math>, <math>[1,2] \mapsto [0,2]</math>, <math>[2,3] \mapsto [2,4] </math>, <math>[3,4] \mapsto [2,4]</math>, etc. This function is the answer to the question: "''If a quantity is measured in a given grid cell of the fine triangulation, which grid cell of the course triangulation will it be measured in?''"

			More generally, the grid cells of the fine triangulation might not be contained inside the grid cells of the coarse triangulation, so a quantity with a given fine-scale measurement might have multiple possible coarse-scale measurements. So really the answer to the italicized question above should not be a function <math>\Gamma' \rightarrow \Gamma </math>, but rather should be a function <math>\Gamma' \rightarrow \text{Prob}(\Gamma) </math>.

Lfox: /* How do we apply functions to continuous quantities? */

2024-10-22T20:31:37Z

How do we apply functions to continuous quantities?

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 == How do we apply functions to continuous quantities? ==
 Let's suppose that we are measuring lengths with a meter stick, which goes down to the nearest 2 millimeters. That is, the ticks on the stick appear at 0mm, 2mm, 4mm, 6mm, etc.
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 Let the grid on one axis consist of intervals <math>[n \pi / 6, (n+1)\pi / 6]</math>, let the grid on the other axis consist of intervals <math>[0.1n, 0.1(n+1)]</math> where <math>n\in \mathbb{Z}</math>, and let the continuum function <math>h(x) = \cos(x)</math>. Then <math>h</math> sends the interval <math>[0, \pi/6]</math> to the interval <math display="block">[\sqrt{3}/2,1] = [0.866\cdots, 1] \subset [0.8, 0.9] \cup [0.9, 1.0]</math>So after applying  <math>h</math> to a length in that internal, we will get a length which is in the interval <math>[0.9, 1.0]</math> with probability <math display="inline">\frac{1}{10 - 5 \sqrt{3}} \approx 0.75 </math>, and in the interval <math>[0.8, 0.9]</math> with probability <math display="inline">\frac{9 - 5 \sqrt{3} }{10 - 5 \sqrt{3}} \approx 0.25 </math>.

Lfox: Created page with "== How do we measure continuous quantities? == First, set up some sort of "triangulation" of the space in which the continuous quantity can take values. For example, a ruler is a "triangulated" line. Typically some sort of quantitative system is chosen to keep track of the cells; e.g. for a ruler we count how many ticks there are. A measurement of a continuous quantity is simply an identification of which one of those intervals the continuous quantity lies inside. It's..."

2024-10-15T21:41:49Z

Created page with "== How do we measure continuous quantities? == First, set up some sort of "triangulation" of the space in which the continuous quantity can take values. For example, a ruler is a "triangulated" line. Typically some sort of quantitative system is chosen to keep track of the cells; e.g. for a ruler we count how many ticks there are. A measurement of a continuous quantity is simply an identification of which one of those intervals the continuous quantity lies inside. It's..."

New page

== How do we measure continuous quantities? ==
First, set up some sort of "triangulation" of the space in which the continuous quantity can take values. For example, a ruler is a "triangulated" line. Typically some sort of quantitative system is chosen to keep track of the cells; e.g. for a ruler we count how many ticks there are.

A measurement of a continuous quantity is simply an identification of which one of those intervals the continuous quantity lies inside. It's a declaration "''this'' quantity lives inside ''this'' interval."

== How do we apply functions to continuous quantities? ==
Let's suppose that we are measuring lengths with a meter stick, which goes down to the nearest 2 millimeters. That is, the ticks on the stick appear at 0mm, 2mm, 4mm, 6mm, etc.

Suppose that we measure a certain length, then, by some process, we double the length. That is, we apply the "continuum" function <math>f(x) = 2x</math>.

That is what <math>f</math> does to length itself, but what does <math>f</math> do to ''measurements'' of length?

Let <math>x</math> denote a length lying somewhere in the range <math>[2k-2, 2k]</math> millimeters (a minimal interval on the meter stick). Then <math>f(x) = 2x</math> lies somewhere in the range <math>[4k-4, 4k] = [4k - 4, 4k -2] \cup [4k - 2, 4k]</math>, which is the union of ''two'' minimal intervals of the meter stick. So maybe we should say that <math>f</math> induces a multi-valued function?

Hmm, but what would be the induced function of <math>g(x) = 1.5x</math>? It would send the interval <math>[2k-2, 2k]</math> to the interval <math>[3k-3, 3k]</math>, which is not necessarily a union of two minimal intervals.

I think it would be best to treat the induced function in a probabilistic manner: If the length <math>x</math> is equally likely to be anywhere in the range <math>[2k-2, 2k]</math>, then

* <math>2x</math> it has a 1/2 chance of being in <math>[4k - 4, 4k -2]</math>, and a 1/2 chance of being in <math>[4k - 2, 4k]</math>.
* <math>3x</math> has,
** if <math>k</math> is even, a 2/3 chance of being in <math>[3k - 2, 3k]</math>, and a 1/3 chance of being in <math>[3k - 4, 3k - 2]</math>.
** if <math>k</math> is odd, a 2/3 chance of being in <math>[3(k+1) - 6, 3(k+1)-4]</math>, and a 1/3 chance of being in <math>[3(k+1) - 4, 3(k+1)-2]</math>.

The general principle is that you have some sort of probability measure on the triangulation, and then you push it forward to another one.

The "problem" with this is that the true probability distributions are "continuum," as the following example demonstrates.

Let the grid on one axis consist of intervals <math>[n \pi / 6, (n+1)\pi / 6]</math>, let the grid on the other axis consist of intervals <math>[0.1n, 0.1(n+1)]</math> where <math>n\in \mathbb{Z}</math>, and let the continuum function <math>h(x) = \cos(x)</math>. Then <math>h</math> sends the interval <math>[0, \pi/6]</math> to the interval <math display="block">[\sqrt{3}/2,1] = [0.866\cdots, 1] \subset [0.8, 0.9] \cup [0.9, 1.0]</math>So after applying <math>h</math> to a length in that internal, we will get a length which is in the interval <math>[0.9, 1.0]</math> with probability <math display="inline">\frac{1}{10 - 5 \sqrt{3}} \approx 0.75 </math>, and in the interval <math>[0.8, 0.9]</math> with probability <math display="inline">\frac{9 - 5 \sqrt{3} }{10 - 5 \sqrt{3}} \approx 0.25 </math>.