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| [[File:Markov Inequality.svg|thumb|300px|right|Markov's inequality gives an upper bound for the measure of the set (indicated in red) where <math>f(x)</math> exceeds a given level <math>\varepsilon</math>. The bound combines the level <math>\varepsilon</math> with the average value of <math>f</math>.]]
| | If your computer is running slow, we have probably gone from the different stages of rage and frustration. Having such a awesome tool like a computer will seem like a curse along with a blessing at the same time whenever this arises. It is good when it's running swiftly and smooth, nevertheless then when it begins acting strange plus slows method down, frustration sets inside. How may something as wonderful because a computer create a individual thus mad?<br><br>Carry out window's program restore. It is important to do this because it removes incorrect changes which have happened in the system. Some of the mistakes result from inability of the program to create restore point regularly.<br><br>If you compare registry cleaners you need a quickly acting registry cleaning. It's no advantageous spending hours and a PC waiting for the registry cleaning to complete its task. We need your cleaner to complete its task in minutes.<br><br>First, constantly clean your PC and keep it free of dust plus dirt. Dirt clogs up all fans plus may result the PC to overheat. You also have to clean up disk area inside order to make a computer run quicker. Delete temporary plus unwanted files plus unused programs. Empty the recycle bin and remove programs we are not utilizing.<br><br>To fix the issue which is caused by registry error, you should use a [http://bestregistrycleanerfix.com/system-mechanic iolo system mechanic]. That is the safest plus simplest technique for average PC users. However there are thousands of registry products available available. You should find a good 1 that really can solve a problem. If you use a terrible 1, you will expect more issues.<br><br>S/w connected error handling - If the blue screen bodily memory dump occurs following the installation of s/w application or perhaps a driver it could be that there is system incompatibility. By booting into safe mode and removing the software you are able to instantly fix this error. You might equally try out a "system restore" to revert to an earlier state.<br><br>Maybe you may be asking why these windows XP error messages appear. Well, for you to be able to know the fix, you need to first recognize where those errors come from. There is this software called registry. A registry is software that shops everything on your PC from a regular configuration, setting, info, and logs of escapades from installing to UN-installing, saving to deleting, along with a lot more alterations you do inside the system pass from it plus gets 'tagged' plus saved because a simple file for recovery reasons. Imagine it as a big recorder, a registrar, of all your records inside your PC.<br><br>Before you buy a complete fresh system; it happens to be time to get the old one cleaned up to start getting more completed online today! Visit our website below and access the most reputable registry cleaner software accessible. |
| In [[probability theory]], '''Markov's inequality''' gives an [[upper bound]] for the [[probability]] that a [[non-negative]] [[function (mathematics)|function]] of a [[random variable]] is greater than or equal to some positive [[Constant (mathematics)|constant]]. It is named after the Russian mathematician [[Andrey Markov]], although it appeared earlier in the work of [[Pafnuty Chebyshev]] (Markov's teacher), and many sources, especially in [[Mathematical analysis|analysis]], refer to it as [[Chebyshev's inequality]] (sometimes, calling it first Chebyshev inequality, while referring to the [[Chebyshev's inequality]] as second Chebyshev's inequality) or [[Irénée-Jules Bienaymé|Bienaymé]]'s inequality.
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| Markov's inequality (and other similar inequalities) relate probabilities to [[expected value|expectation]]s, and provide (frequently loose but still useful) bounds for the [[cumulative distribution function]] of a random variable.
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| An example of an application of Markov's inequality is the fact that (assuming incomes are non-negative) no more than 1/5 of the population can have more than 5 times the average income.
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| ==Statement==
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| If ''X'' is any nonnegative random variable and ''a'' > 0, then
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| : <math>\mathbb{P}(X \geq a) \leq \frac{\mathbb{E}(X)}{a}.</math>
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| In the language of [[measure theory]], Markov's inequality states that if (''X'', Σ, ''μ'') is a [[Measure (mathematics)|measure space]], ''ƒ'' is a [[measurable function|measurable]] [[extended real number line|extended real]]-valued function, and <math>\varepsilon>0</math>, then
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| :<math> \mu(\{x\in X:|f(x)|\geq \varepsilon \}) \leq {1\over \varepsilon}\int_X |f|\,d\mu.</math>
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| (This measure theoretic definition may sometimes be referred to as [[Chebyshev's inequality]]
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| .<ref>E.M. Stein, R. Shakarchi, "Real Analysis, Measure Theory, Integration, & Hilbert Spaces", vol. 3, 1st ed., 2005, p.91</ref>) | |
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| ==Corollary: Chebyshev's inequality==
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| [[Chebyshev's inequality]] uses the variance to bound the probability that a random variable deviates far from the mean. Specifically:
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| :<math>\mathbb{P}(|X-\mathbb{E}(X)| \geq a) \leq \frac{\textrm{Var}(X)}{a^2},</math>
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| for any ''a>0''. Here Var(X) is the variance of X, defined as:
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| :<math> \operatorname{Var}(X) = \mathbb{E}[(X - \mathbb{E}(X) )^2]. </math>
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| Chebyshev's inequality follows from Markov's inequality by considering the random variable
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| : <math> (X - \mathbb{E}(X))^2 </math>
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| for which Markov's inequality reads
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| : <math> \mathbb{P}( (X - \mathbb{E}(X))^2 \ge a^2) \le \frac{\operatorname{Var}(X)}{a^2}, </math>
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| ==Proofs==
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| We separate the case in which the measure space is a probability space from the more general case because the probability case is more accessible for the general reader.
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| ===Proof In the language of probability theory===
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| For any event ''E'', let ''I''<sub>''E''</sub> be the indicator random variable of ''E'', that is, ''I''<sub>''E''</sub> = 1 if ''E'' occurs and ''I''<sub>''E''</sub> = 0 otherwise.
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| Using this notation, we have ''I''<sub>(''X'' ≥ ''a'')</sub> = 1 if the event ''X'' ≥ ''a'' occurs, and ''I''<sub>(''X'' ≥ ''a'')</sub> = 0 if ''X'' < ''a''. Then, given ''a'' > 0,
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| :<math>aI_{(X \geq a)} \leq X\,</math>
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| which is clear if we consider the two possible values of ''I''<sub>(''X'' ≥ ''a'')</sub>. If ''X'' < ''a'', then ''I''<sub>(''X'' ≥ ''a'')</sub> = 0, and so ''aI''<sub>(''X'' ≥ ''a'')</sub> = 0 ≤ ''X''. Otherwise, we have ''X'' ≥ ''a'', for which ''I''<sub>(''X'' ≥ ''a'')</sub> = 1 and so ''aI''<sub>(''X'' ≥ ''a'')</sub> = a ≤ ''X''.
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| Since <math>\mathbb{E}</math> is a linear operator, taking expectation of both sides of an inequality cannot reverse it. Therefore
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| :<math>\mathbb{E}(aI_{(X \geq a)}) \leq \mathbb{E}(X).\,</math>
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| Now, using linearity of expectations, the left side of this inequality is the same as
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| :<math>a\mathbb{E}(I_{(X \geq a)}) = a(1\cdot\mathbb{P}(X \geq a) + 0\cdot\mathbb{P}(X < a)) = a\mathbb{P}(X \geq a).\,</math>
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| Thus we have
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| :<math>a\mathbb{P}(X \geq a) \leq \mathbb{E}(X)\,</math>
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| and since ''a'' > 0, we can divide both sides by ''a''. | |
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| ===In the language of measure theory===
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| We may assume that the function <math>f</math> is non-negative, since only its absolute value enters in the equation. Now, consider the real-valued function ''s'' on ''X'' given by
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| :<math>
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| s(x) =
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| \begin{cases}
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| \varepsilon, & \text{if } f(x) \geq \varepsilon \\
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| 0, & \text{if } f(x) < \varepsilon
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| \end{cases}
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| </math>
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| Then <math>0\leq s(x)\leq f(x)</math>. By the definition of the [[Lebesgue integral]]
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| :<math>
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| \int_X f(x) \, d\mu \geq \int_X s(x) \, d \mu = \varepsilon \mu( \{ x\in X : \, f(x) \geq \varepsilon \} )
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| </math>
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| and since <math>\varepsilon >0 </math>, both sides can be divided by <math>\varepsilon</math>, obtaining
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| :<math>\mu(\{x\in X : \, f(x) \geq \varepsilon \}) \leq {1\over \varepsilon }\int_X f \,d\mu.</math>
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| [[Q.E.D.]]
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| ==Matrix-valued Markov==
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| Let <math> M \succeq 0 </math> be a self adjoint matrix-valued random variable and <math> a>0 </math>. Then
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| :<math>
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| \mathbb{P}(M \npreceq a \cdot I) \leq \frac{\mathrm{tr}\left( E(M) \right)}{a}.
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| </math>
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| ==Examples==
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| * Markov's inequality is used to prove [[Chebyshev's inequality]].
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| * Markov's inequality can be used to show that, for a nonnegative random variable, the mean <math>\mu</math> and a median <math>m</math> are such that <math>m \le 2 \mu</math>.
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| ==See also==
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| * [[Doob martingale#McDiarmid.27s_inequality|McDiarmid's inequality]]
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| * [[Bernstein inequalities (probability theory)]]
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| *[[Chernoff_bound|Chernoff Bound]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| * [http://mws.cs.ru.nl/mwiki/random_1.html#T36 The formal proof of Markov's inequality] in the [[Mizar system]].
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| {{Refimprove|date=September 2010}}
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| {{DEFAULTSORT:Markov's Inequality}}
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| [[Category:Probabilistic inequalities]]
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| [[Category:Articles containing proofs]]
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If your computer is running slow, we have probably gone from the different stages of rage and frustration. Having such a awesome tool like a computer will seem like a curse along with a blessing at the same time whenever this arises. It is good when it's running swiftly and smooth, nevertheless then when it begins acting strange plus slows method down, frustration sets inside. How may something as wonderful because a computer create a individual thus mad?
Carry out window's program restore. It is important to do this because it removes incorrect changes which have happened in the system. Some of the mistakes result from inability of the program to create restore point regularly.
If you compare registry cleaners you need a quickly acting registry cleaning. It's no advantageous spending hours and a PC waiting for the registry cleaning to complete its task. We need your cleaner to complete its task in minutes.
First, constantly clean your PC and keep it free of dust plus dirt. Dirt clogs up all fans plus may result the PC to overheat. You also have to clean up disk area inside order to make a computer run quicker. Delete temporary plus unwanted files plus unused programs. Empty the recycle bin and remove programs we are not utilizing.
To fix the issue which is caused by registry error, you should use a iolo system mechanic. That is the safest plus simplest technique for average PC users. However there are thousands of registry products available available. You should find a good 1 that really can solve a problem. If you use a terrible 1, you will expect more issues.
S/w connected error handling - If the blue screen bodily memory dump occurs following the installation of s/w application or perhaps a driver it could be that there is system incompatibility. By booting into safe mode and removing the software you are able to instantly fix this error. You might equally try out a "system restore" to revert to an earlier state.
Maybe you may be asking why these windows XP error messages appear. Well, for you to be able to know the fix, you need to first recognize where those errors come from. There is this software called registry. A registry is software that shops everything on your PC from a regular configuration, setting, info, and logs of escapades from installing to UN-installing, saving to deleting, along with a lot more alterations you do inside the system pass from it plus gets 'tagged' plus saved because a simple file for recovery reasons. Imagine it as a big recorder, a registrar, of all your records inside your PC.
Before you buy a complete fresh system; it happens to be time to get the old one cleaned up to start getting more completed online today! Visit our website below and access the most reputable registry cleaner software accessible.