Morten Hjorth-Jensen
463 lines
26 KiB
HTML
463 lines
26 KiB
HTML
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<!-- navigation toc: --> <li><a href="._Statistics-bs001.html#___sec0" style="font-size: 80%;"> Domains and probabilities</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs002.html#___sec1" style="font-size: 80%;"> Tossing a dice</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs003.html#___sec2" style="font-size: 80%;"> Stochastic variables</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs004.html#___sec3" style="font-size: 80%;"> Stochastic variables and the main concepts, the discrete case</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs005.html#___sec4" style="font-size: 80%;"> Stochastic variables and the main concepts, the continuous case</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs006.html#___sec5" style="font-size: 80%;"> The cumulative probability</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs007.html#___sec6" style="font-size: 80%;"> Properties of PDFs</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs008.html#___sec7" style="font-size: 80%;"> Important distributions, the uniform distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs009.html#___sec8" style="font-size: 80%;"> Gaussian distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs010.html#___sec9" style="font-size: 80%;"> Exponential distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs011.html#___sec10" style="font-size: 80%;"> Expectation values</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs012.html#___sec11" style="font-size: 80%;"> Stochastic variables and the main concepts, mean values</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs013.html#___sec12" style="font-size: 80%;"> Stochastic variables and the main concepts, central moments, the variance</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs014.html#___sec13" style="font-size: 80%;"> Probability Distribution Functions</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs015.html#___sec14" style="font-size: 80%;"> Probability Distribution Functions</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs016.html#___sec15" style="font-size: 80%;"> The three famous Probability Distribution Functions</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs017.html#___sec16" style="font-size: 80%;"> Probability Distribution Functions, the normal distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs018.html#___sec17" style="font-size: 80%;"> Probability Distribution Functions, the normal distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs019.html#___sec18" style="font-size: 80%;"> Probability Distribution Functions, the cumulative distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs020.html#___sec19" style="font-size: 80%;"> Probability Distribution Functions, other important distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs021.html#___sec20" style="font-size: 80%;"> Probability Distribution Functions, the binomial distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs022.html#___sec21" style="font-size: 80%;"> Probability Distribution Functions, Poisson's distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs023.html#___sec22" style="font-size: 80%;"> Probability Distribution Functions, Poisson's distribution</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs024.html#___sec23" style="font-size: 80%;"> Meet the covariance!</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs025.html#___sec24" style="font-size: 80%;"> Meet the covariance in matrix disguise</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs026.html#___sec25" style="font-size: 80%;"> Meet the covariance, uncorrelated events</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs027.html#___sec26" style="font-size: 80%;"> Numerical experiments and the covariance</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs028.html#___sec27" style="font-size: 80%;"> Numerical experiments and the covariance</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs029.html#___sec28" style="font-size: 80%;"> Numerical experiments and the covariance, actual situations</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs030.html#___sec29" style="font-size: 80%;"> Numerical experiments and the covariance, our observables</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs031.html#___sec30" style="font-size: 80%;"> Numerical experiments and the covariance, the sample variance</a></li>
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<!-- navigation toc: --> <li><a href="#___sec31" style="font-size: 80%;"> Numerical experiments and the covariance, central limit theorem</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs033.html#___sec32" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs034.html#___sec33" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs035.html#___sec34" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs036.html#___sec35" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs037.html#___sec36" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation, sample variance</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs038.html#___sec37" style="font-size: 80%;"> Definition of Correlation Functions and Standard Deviation</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs039.html#___sec38" style="font-size: 80%;"><b>Random Numbers</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs040.html#___sec39" style="font-size: 80%;"><b>Random Numbers, better name: pseudo random numbers</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs041.html#___sec40" style="font-size: 80%;"><b>Random number generator RNG</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs042.html#___sec41" style="font-size: 80%;"><b>Random number generator RNG and periodic outputs</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs043.html#___sec42" style="font-size: 80%;"><b>Random number generator RNG and its period</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs044.html#___sec43" style="font-size: 80%;"><b>Random number generator RNG, other examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs045.html#___sec44" style="font-size: 80%;"><b>Random number generator RNG, other examples</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs046.html#___sec45" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs047.html#___sec46" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs048.html#___sec47" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs049.html#___sec48" style="font-size: 80%;"><b>Random number generator RNG, RAN0</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs050.html#___sec49" style="font-size: 80%;"><b>Random number generator RNG, RAN0 code</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs051.html#___sec50" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs052.html#___sec51" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs053.html#___sec52" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs054.html#___sec53" style="font-size: 80%;"> Simple demonstration of RNGs using python</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs055.html#___sec54" style="font-size: 80%;"> Properties of Selected Random Number Generators</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs056.html#___sec55" style="font-size: 80%;"> Correlation function and which random number generators should I use</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs057.html#___sec56" style="font-size: 80%;"> Correlation function and which random number generators should I use</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs058.html#___sec57" style="font-size: 80%;"><b>Which RNG should I use?</b></a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs059.html#___sec58" style="font-size: 80%;"> How to use the Mersenne generator</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs060.html#___sec59" style="font-size: 80%;"> Why blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs061.html#___sec60" style="font-size: 80%;"> Why blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs062.html#___sec61" style="font-size: 80%;"> Code to demonstrate the calculation of the autocorrelation function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs063.html#___sec62" style="font-size: 80%;"> What is blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs064.html#___sec63" style="font-size: 80%;"> What is blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs065.html#___sec64" style="font-size: 80%;"> What is blocking?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._Statistics-bs066.html#___sec65" style="font-size: 80%;"> Implementation</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs067.html#___sec66" style="font-size: 80%;"> Actual implementation with code, main function</a></li>
|
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<!-- navigation toc: --> <li><a href="._Statistics-bs068.html#___sec67" style="font-size: 80%;"> The Bootstrap method</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs069.html#___sec68" style="font-size: 80%;"> Bootstrapping</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs070.html#___sec69" style="font-size: 80%;"> Bootstrapping, recipe</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs071.html#___sec70" style="font-size: 80%;"> Bootstrapping, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py"</a></li>
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<!-- navigation toc: --> <li><a href="._Statistics-bs072.html#___sec71" style="font-size: 80%;"> Jackknife, "code":"https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py"</a></li>
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<a name="part0032"></a>
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<!-- !split -->
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<h2 id="___sec31" class="anchor">Numerical experiments and the covariance, central limit theorem </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>
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The central limit theorem states that the PDF \( \tilde{p}(z) \) of
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the average of \( m \) random values corresponding to a PDF \( p(x) \)
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is a normal distribution whose mean is the
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mean value of the PDF \( p(x) \) and whose variance is the variance
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of the PDF \( p(x) \) divided by \( m \), the number of values used to compute \( z \).
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<p>
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The central limit theorem leads then to the well-known expression for the
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standard deviation, given by
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$$
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\begin{equation*}
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\sigma_m=
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\frac{\sigma}{\sqrt{m}}.
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\end{equation*}
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$$
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<p>
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In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results.
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