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{{Unreferenced|date=March 2011}}
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{{Probability distribution
| name      = Cantor
| type      = mass <!-- not technically correct; added only so template pmf entry would parse correctly; used "mass" rather than "density" since cdf plot suggests it *might* have a pmf -->
| parameters = none
| support    = [[Cantor set]]
| pdf        = none
| cdf        = [[Cantor function]]
| mean      = 1/2
| median    = anywhere in [1/3,&nbsp;2/3]
| mode      = n/a
| variance  = 1/8
| skewness  = 0
| kurtosis  = &minus;8/5
| entropy    =
| mgf        = <math>e^{t/2}
                  \prod_{i= 1}^{\infty} \cosh{\left(\frac{t}{3^{i}}
                                              \right)}</math>
| char      = <math>e^{\mathrm{i}\,t/2}
                  \prod_{i= 1}^{\infty} \cos{\left(\frac{t}{3^{i}}
                                              \right)}</math>
}}
 
The '''Cantor distribution''' is the [[probability distribution]] whose [[cumulative distribution function]] is the '''[[Cantor function]]'''.
 
This distribution has neither a [[probability density function]] nor a [[probability mass function]], as it is not [[absolute continuity|absolutely continuous]] with respect to [[Lebesgue measure]], nor has it any point-masses. It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these. Rather it is an example of a [[singular distribution]].
 
Its cumulative distribution function is sometimes referred to as the [[Singular function|Devil's staircase]], although that term has a more general meaning.
 
== Characterization ==
 
The [[Support (mathematics)|support]] of the Cantor distribution is the [[Cantor set]], itself the intersection of the (countably infinitely many) sets
 
:<math>
\begin{align}
C_{0} = & [0,1] \\
C_{1} = & [0,1/3]\cup[2/3,1] \\
C_{2} = & [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1] \\
C_{3} = & [0,1/27]\cup[2/27,1/9]\cup[2/9,7/27]\cup[8/27,1/3]\cup \\
        & [2/3,19/27]\cup[20/27,7/9]\cup[8/9,25/27]\cup[26/27,1] \\
C_{4} = & \cdots .
\end{align}
</math>
 
The Cantor distribution is the unique probability distribution for which for any ''C''<sub>''t''</sub> (''t''&nbsp;∈&nbsp;{&nbsp;0,&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...&nbsp;}), the probability of a particular interval in ''C''<sub>''t''</sub> containing the Cantor-distributed random variable is identically 2<sup>-''t''</sup> on each one of the 2<sup>''t''</sup> intervals.
 
== Moments ==
It is easy to see by symmetry that for a [[random variable]] ''X'' having this distribution, its [[expected value]] E(''X'') = 1/2, and that all odd central moments of ''X'' are 0.
 
The [[law of total variance]] can be used to find the [[variance]] var(''X''), as follows.  For the above set ''C''<sub>1</sub>, let ''Y'' = 0 if ''X''&nbsp;∈&nbsp;[0,1/3], and 1 if ''X''&nbsp;∈&nbsp;[2/3,1]. Then:
 
: <math>
\begin{align}
\operatorname{var}(X) & = \operatorname{E}(\operatorname{var}(X\mid Y)) +
                          \operatorname{var}(\operatorname{E}(X\mid Y)) \\
                      & = \frac{1}{9}\operatorname{var}(X) +
                          \operatorname{var}
                            \left\{
                            \begin{matrix} 1/6 & \mbox{with probability}\ 1/2 \\
                                            5/6 & \mbox{with probability}\ 1/2
                            \end{matrix}
                            \right\} \\
                      & = \frac{1}{9}\operatorname{var}(X) + \frac{1}{9}
\end{align}
</math>
 
From this we get:
 
:<math>\operatorname{var}(X)=\frac{1}{8}.</math>
 
A closed-form expression for any even [[central moment]] can be found by first obtaining the even [[cumulants]][http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf]
 
:<math>
\kappa_{2n} = \frac{2^{2n-1} (2^{2n}-1) B_{2n}}
                    {n\, (3^{2n}-1)}, \,\!
</math>
 
where ''B''<sub>2''n''</sub> is the 2''n''th [[Bernoulli number]], and then [[Cumulant#Cumulants and moments|expressing the moments as functions of the cumulants]].
 
== References ==
{{Reflist}}
 
== External links ==
 
* {{cite web |last=Morrison |first=Kent |url=http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf |title=Random Walks with Decreasing Steps |publisher=Department of Mathematics, California Polytechnic State University |date=1998-07-23 |accessdate=2007-02-16}}
 
{{ProbDistributions|miscellaneous}}
{{-}}
 
[[Category:Probability distributions]]

Latest revision as of 22:46, 25 September 2014

The following post is about 5 important procedures that might guide you inside your pursuit of getting slim plus usually coach we on how to get rid of 70 pounds inside simply 2 months and all right factors to do. If you apply these steps daily, you are able to lose weight plus receive to the actual weight we desire.

So when bmr calculator this is happening to you and you don't wish To be apple shaped, what will you do about it. First, consider the diet. Cut out processed foods and saturated fat. Items like butter, cakes, biscuits, full fat milk, white bread, willing prepared food.

MyFitnessPal attributes a food diary, exercise log, online forums, and blog with helpful fat reduction strategies. It equally offers calculators for your heart rate, body mass index (BMI), basal metabolic rate (BMR), plus various dietary values.

11) Simply as we just must eat while you're hungry, only sip on sweet-tasting liquids when you're hungry. Consider everything you put inside your mouth with the exception of water to be food, thus focus on the right food, balanced food throughout the course of the day.

How do you know how countless calories we require? There is not any magic amount that fits everybody. Many factors have a character in determining what the appropriate amount is for you. Some of these factors include age, present fat, activity level, height, and bmr.

To calculate your BMR, try using this free online calculator: http://www.caloriesperhour.com/index_burn.html . When we have the results, jot them down on a piece of paper for reference.

Tabata training is a quite significant intensity shape of exercise. For which reason your anaerobic and aerobic abilities might increase. Aerobics is the ability for the body to take in because much oxygen because needed and circulate it to the muscles of your body. However, when you may be moving too fast where the oxygen has no time to circulate, then we begin to make energy anaerobically with glucose. In this means tabata training usually moreover heighten endurance.

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