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| | This is a quite well-known frustration amidst those of we whom are desperate to get rid of a few pounds. You sincerely believe that you are doing everything right, however the scale stubbornly refuses to budge. What's going on?<br><br>Whenever you do the variable wog we just change a speed of movement intermittently. We walk for a while, we walk for a while, we calorie burn calculator jog for a while, and we steady wog for a while. You get exercise while you enjoy oneself.<br><br>Let's consider what a standard calorie counter would have performed. First, it can assume a typical calorie burn rate of 1 calorie per minute. Then the counter usually find which exercising for 30 minutes can yield 30 (minutes) * 6 (MET) * 1 (calories per minute) = 180 calories. The calorie counter might add these 180 calories to your daily expenditure without because a part of these 180 calories is absolutely accounted by a routine escapades.<br><br>For instance if a 200 lb man desired to lose 20 lbs Id recommend he shoot for doing it in [http://safedietplansforwomen.com/calories-burned-walking calorie burn calculator] 2 months time period. He can lose 1-2 lbs securely each week while furthermore maintaining his muscle mass.<br><br>If you intensify your workout, you are able to continuously burn fat. After you exercise your calories burned calculator metabolism speeds up. Thus, we should exercise each morning to optimize the effects of burning fats. Consult our significant source about body fat calculator by clicking the link.<br><br>Many folks count the calories they spend exercising because "extra" calories. There is a difference between calories burned while exercising and "extra" calories burned exercising. Here is an example: we burn 300 calories on the treadmill instead of the normal activity (viewing TV at home); in reality, you have to subtract the calories you'd have spent viewing TV from these 300 calories to calculate how countless more calories you burned. Let's say that watching TV, you'd have burned 80 calories. In this specific case, we have expended 300 calories while exercising, plus 220 "extra" calories.<br><br>The key to not dropping prey to such supermarket schemes is 'mind control'. It's all inside the notice! Keep telling yourself you're not going to succumb to any temptation. If you win the battle inside the notice, we are sure to have self control inside the mall. Being certain regarding what you need, might assist avoid indulging in impulse obtaining. |
| [[File:Normal Distribution CDF.svg|thumb|300px|Cumulative distribution function for the normal distributions in the image below.]]
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| [[File:Normal Distribution PDF.svg|thumb|300px|Probability density function for several normal distributions. The red line denotes the standard normal distribution.]]
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| In [[probability theory]] and [[statistics]], the '''cumulative distribution function''' ('''CDF'''), or just '''distribution function''', describes the probability that a real-valued [[random variable]] ''X'' with a given [[probability distribution]] will be found at a value less than or equal to ''x''. In the case of a [[continuous distribution]], it gives the area under the [[probability density function]] from minus infinity to ''x''. Cumulative distribution functions are also used to specify the distribution of [[multivariate random variable]]s.
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| ==Definition==
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| The cumulative distribution function of a real-valued [[random variable]] ''X'' is the function given by
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| :<math>F_X(x) = \operatorname{P}(X\leq x),</math>
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| where the right-hand side represents the [[probability]] that the random variable ''X'' takes on a value less than or
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| equal to ''x''. The probability that ''X'' lies in the semi-closed [[interval (mathematics)|interval]] (''a'', ''b''<nowiki>]</nowiki>, where ''a'' < ''b'', is therefore
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| :<math>\operatorname{P}(a < X \le b)= F_X(b)-F_X(a).</math>
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| In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but is important for discrete distributions. The proper use of tables of the [[Binomial distribution|binomial]] and [[Poisson distribution]]s depends upon this convention. Moreover, important formulas like [[Paul Lévy (mathematician)|Paul Lévy]]'s inversion formula for the [[Characteristic function (probability theory)#Inversion formulas|characteristic function]] also rely on the "less than or equal" formulation.
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| If treating several random variables ''X'', ''Y'', ... etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital ''F'' for a cumulative distribution function, in contrast to the lower-case ''f'' used for [[probability density function]]s and [[probability mass function]]s. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the [[normal distribution]].
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| The CDF of a [[continuous random variable]] ''X'' can be expressed as the integral of its [[probability density function]] ƒ<sub>X</sub> as follows:
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| :<math>F_X(x) = \int_{-\infty}^x f_X(t)\,dt.</math>
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| In the case of a random variable ''X'' which has distribution having a discrete component at a value ''b'',
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| :<math>\operatorname{P}(X=b) = F_X(b) - \lim_{x \to b^{-}} F_X(x).</math>
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| If ''F<sub>X</sub>'' is continuous at ''b'', this equals zero and there is no discrete component at ''b''.
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| == Properties ==
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| [[Image:Discrete probability distribution illustration.svg|right|thumb|From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part.]]
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| Every cumulative distribution function ''F'' is [[monotone increasing|non-decreasing]] and [[right-continuous]], which makes it a [[càdlàg]] function. Furthermore,
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| :<math>\lim_{x\to -\infty}F(x)=0, \quad \lim_{x\to +\infty}F(x)=1.</math>
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| Every function with these four properties is a CDF, i.e., for every such function, a [[random variable]] can be defined such that the function is the cumulative distribution function of that random variable.
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| If ''X'' is a purely [[discrete random variable]], then it attains values ''x''<sub>1</sub>, ''x''<sub>2</sub>, ... with probability ''p''<sub>i</sub> = P(''x''<sub>i</sub>), and the CDF of ''X'' will be discontinuous at the points ''x''<sub>''i''</sub> and constant in between:
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| :<math>F(x) = \operatorname{P}(X\leq x) = \sum_{x_i \leq x} \operatorname{P}(X = x_i) = \sum_{x_i \leq x} p(x_i).</math>
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| If the CDF ''F'' of ''X'' is [[continuous function|continuous]], then ''X'' is a [[continuous random variable]]; if furthermore ''F'' is [[absolute continuity|absolutely continuous]], then there exists a [[Lebesgue integral|Lebesgue-integrable]] function ''f''(''x'') such that
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| :<math>F(b)-F(a) = \operatorname{P}(a< X\leq b) = \int_a^b f(x)\,dx</math>
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| for all real numbers ''a'' and ''b''. The function ''f'' is equal to the [[derivative]] of ''F'' [[almost everywhere]], and it is called the [[probability density function]] of the distribution of ''X''.
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| == Examples ==
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| As an example, suppose ''X'' is [[uniform distribution (continuous)|uniformly distributed]] on the unit interval [0, 1].
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| Then the CDF of X is given by
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| :<math>F(x) = \begin{cases} | |
| 0 &:\ x < 0\\
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| x &:\ 0 \le x < 1\\
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| 1 &:\ 1 \le x.
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| \end{cases}</math>
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| Suppose instead that ''X'' takes only the discrete values 0 and 1, with equal probability.
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| Then the CDF of X is given by
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| :<math>F(x) = \begin{cases}
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| 0 &:\ x < 0\\
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| 1/2 &:\ 0 \le x < 1\\ | |
| 1 &:\ 1 \le x.
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| \end{cases}</math>
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| ==Derived functions==
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| ===Complementary cumulative distribution function (tail distribution)===<!-- This section is linked from [[Power law]], [[Stretched exponential function]] and [[Weibull distribution]] -->
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| Sometimes, it is useful to study the opposite question and ask how often the random variable is ''above'' a particular level. This is called the '''complementary cumulative distribution function''' ('''ccdf''') or simply the '''tail distribution''' or '''exceedance''', and is defined as
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| :<math>\bar F(x) = \operatorname{P}(X > x) = 1 - F(x).</math>
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| This has applications in [[statistics|statistical]] [[hypothesis test]]ing, for example, because the one-sided [[p-value]] is the probability of observing a test statistic ''at least'' as extreme as the one observed. Thus, provided that the [[test statistic]], ''T'', has a continuous distribution, the one-sided [[p-value]] is simply given by the ccdf: for an observed value ''t'' of the test statistic
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| :<math>p= \operatorname{P}(T \ge t) = \operatorname{P}(T > t) =1 - F_T(t).</math>
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| In [[survival analysis]], <math>\bar F(x)</math> is called the '''[[survival function]]''' and denoted <math> S(x) </math>, while the term ''reliability function'' is common in [[engineering]].
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| ;Properties
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| * For a non-negative continuous random variable having an expectation, [[Markov's inequality]] states that<ref name="ZK">{{cite book| last1 = Zwillinger| first1 = Daniel| last2 = Kokoska| first2 = Stephen| title = CRC Standard Probability and Statistics Tables and Formulae| year = 2010| publisher = CRC Press| isbn = 978-1-58488-059-2| page = 49 }}</ref>
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| :: <math>\bar F(x) \leq \frac{\mathbb E(X)}{x} .</math>
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| * As <math> x \to \infty, \bar F(x) \to 0 \ </math>, and in fact <math> \bar F(x) = o(1/x) .</math>
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| :Proof:{{citation needed|date=April 2012}} Assuming X has a density function f, for any <math> c> 0 </math>
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| ::<math>
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| \mathbb E(X) = \int_0^\infty xf(x)dx \geq \int_0^c xf(x)dx + c\int_c^\infty f(x)dx
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| </math>
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| :Then, on recognizing <math>\bar F(c) = \int_c^\infty f(x)dx </math> and rearranging terms,
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| ::<math>
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| 0 \leq c\bar F(c) \leq \mathbb E(X) - \int_0^c x f(x)dx \to 0 \text{ as } c \to \infty
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| </math>
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| :as claimed.
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| ===Folded cumulative distribution===
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| [[Image:Folded-cumulative-distribution-function.svg|thumb|right|Example of the folded cumulative distribution for a [[normal distribution]] function with an [[expected value]] of 0 and a [[standard deviation]] of 1.]]
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| While the plot of a cumulative distribution often has an S-like shape, an alternative illustration is the '''folded cumulative distribution''' or '''mountain plot''', which folds the top half of the graph over,<ref name="Gentle">{{cite book| author = Gentle, J.E.| title = Computational Statistics| url = http://books.google.com/?id=m4r-KVxpLsAC&pg=PA348| accessdate = 2010-08-06| year = 2009| publisher = [[Springer Science+Business Media|Springer]]| isbn = 978-0-387-98145-1 }}{{Page needed|date=June 2011}}</ref><ref name="Monti">
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| {{cite journal|author=Monti, K.L.|pages=342–345|year=1995|title=Folded Empirical Distribution Function Curves (Mountain Plots) |journal=The American Statistician|volume=49|jstor=2684570}}</ref>
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| thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the [[median (statistics)|median]] and [[dispersion (statistics)|dispersion]] (the [[mean absolute deviation]] from the median<ref>{{Cite journal
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| | last1 = Xue | first1 = J. H.
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| | last2 = Titterington | first2 = D. M.
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| | doi = 10.1016/j.spl.2011.03.014
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| | title = The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile
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| | journal = Statistics & Probability Letters
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| | volume = 81 | issue = 8 | pages = 1179–1182
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| | year = 2011
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| | pmid = | pmc = }}<</ref>) of the distribution or of the empirical results.
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| ===Inverse distribution function (quantile function)===
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| If the CDF ''F'' is strictly increasing and continuous then <math> F^{-1}( y ), y \in [0,1] </math> is the unique real number <math> x </math> such that <math> F(x) = y </math>. In such a case, this defines the '''inverse distribution function''' or [[quantile function]].
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| Unfortunately, the distribution does not, in general, have an inverse. One may define, for <math> y \in [0,1] </math>, the '''generalized inverse distribution function''':
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| :<math>
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| F^{-1}(y) = \inf_{x \in \mathbb{R}} \{ F(x) \geq y \}.
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| </math>
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| * Example 1: The median is <math>F^{-1}( 0.5 )</math>.
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| * Example 2: Put <math> \tau = F^{-1}( 0.95 ) </math>. Then we call <math> \tau </math> the 95th percentile.
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| The inverse of the cdf is called the [[quantile function]]. | |
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| The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions. Some useful properties of the inverse cdf are:
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| # <math>F^{-1}</math> is nondecreasing
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| # <math>F^{-1}(F(x)) \leq x</math>
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| # <math>F(F^{-1}(y)) \geq y</math>
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| # <math>F^{-1}(y) \leq x</math> if and only if <math>y \leq F(x)</math>
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| # If <math>Y</math> has a <math>U[0, 1]</math> distribution then <math>F^{-1}(Y)</math> is distributed as <math>F</math>. This is used in [[random number generation]] using the [[inverse transform sampling]]-method.
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| # If <math>\{X_\alpha\}</math> is a collection of independent <math>F</math>-distributed random variables defined on the same sample space, then there exist random variables <math>Y_\alpha</math> such that <math>Y_\alpha</math> is distributed as <math>U[0,1]</math> and <math>F^{-1}(Y_\alpha) = X_\alpha</math> with probability 1 for all <math>\alpha</math>.
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| ==Multivariate case ==
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| When dealing simultaneously with more than one random variable the ''joint'' cumulative distribution function can also be defined. For example, for a pair of random variables ''X,Y'', the joint CDF <math>F</math> is given by
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| :<math>F(x,y) = \operatorname{P}(X\leq x,Y\leq y),</math>
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| where the right-hand side represents the [[probability]] that the random variable ''X'' takes on a value less than or
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| equal to ''x'' and that ''Y'' takes on a value less than or
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| equal to ''y''.
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| Every multivariate CDF is:
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| # Monotonically non-decreasing for each of its variables
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| # Right-continuous for each of its variables.
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| # <math>0\leq F(x_{1},...,x_{n})\leq 1</math>
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| # <math>\lim_{x_{1},...,x_{n}\rightarrow+\infty}F(x_{1},...,x_{n})=1</math> and <math>\lim_{x_{i}\rightarrow-\infty}F(x_{1},...,x_{n})=0,\quad \mbox{for all i}</math>
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| ==Use in statistical analysis==
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| The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. [[Cumulative frequency analysis]] is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The [[empirical distribution function]] is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various [[statistical hypothesis test]]s. Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution, or evidence against two samples of data having arisen from the same (unknown) population distribution.
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| ===Kolmogorov–Smirnov and Kuiper's tests===
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| The [[Kolmogorov–Smirnov test]] is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related [[Kuiper's test]] is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.
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| ==See also==
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| * [[Descriptive statistics]]
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| * [[Distribution fitting]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{commons category-inline|Cumulative distribution functions}}
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| {{Theory of probability distributions}}
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| {{DEFAULTSORT:Cumulative Distribution Function}}
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| [[Category:Theory of probability distributions]]
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This is a quite well-known frustration amidst those of we whom are desperate to get rid of a few pounds. You sincerely believe that you are doing everything right, however the scale stubbornly refuses to budge. What's going on?
Whenever you do the variable wog we just change a speed of movement intermittently. We walk for a while, we walk for a while, we calorie burn calculator jog for a while, and we steady wog for a while. You get exercise while you enjoy oneself.
Let's consider what a standard calorie counter would have performed. First, it can assume a typical calorie burn rate of 1 calorie per minute. Then the counter usually find which exercising for 30 minutes can yield 30 (minutes) * 6 (MET) * 1 (calories per minute) = 180 calories. The calorie counter might add these 180 calories to your daily expenditure without because a part of these 180 calories is absolutely accounted by a routine escapades.
For instance if a 200 lb man desired to lose 20 lbs Id recommend he shoot for doing it in calorie burn calculator 2 months time period. He can lose 1-2 lbs securely each week while furthermore maintaining his muscle mass.
If you intensify your workout, you are able to continuously burn fat. After you exercise your calories burned calculator metabolism speeds up. Thus, we should exercise each morning to optimize the effects of burning fats. Consult our significant source about body fat calculator by clicking the link.
Many folks count the calories they spend exercising because "extra" calories. There is a difference between calories burned while exercising and "extra" calories burned exercising. Here is an example: we burn 300 calories on the treadmill instead of the normal activity (viewing TV at home); in reality, you have to subtract the calories you'd have spent viewing TV from these 300 calories to calculate how countless more calories you burned. Let's say that watching TV, you'd have burned 80 calories. In this specific case, we have expended 300 calories while exercising, plus 220 "extra" calories.
The key to not dropping prey to such supermarket schemes is 'mind control'. It's all inside the notice! Keep telling yourself you're not going to succumb to any temptation. If you win the battle inside the notice, we are sure to have self control inside the mall. Being certain regarding what you need, might assist avoid indulging in impulse obtaining.