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Normal (Gaussian) distribution 

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\(f_X(x; \mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}exp \left(-\frac{(x-\mu) ^2}{2\sigma^2}\right)=\mathcal{N}(\mu,\sigma)\)

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What is intermittency?

A signal is said to be intermittent if rare events of large magnitude are separated by long periods with events of low magnitude. Spatial intermittency implies that the signal displays localized regions with events of large magnitude, and wide areas with events of low magnitude. 

  • PDFs of intermittent flows are not Gaussian. 
  • Kurtosis is often used as a measure of intermittency (high intermittency means high kurtosis) 
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Can the variance be zero? 

Yes, then:

  • the distribution only consists of one constant 
  • mean, median and mode are the same 
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Tell me a distribution where no moments exist:

The Cauchy distribution 

  • expected value, variance and standard deviation do not exist since it's integrals are infinite. 

 

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Law of large numbers 

Given a sequence of random variables \(X_1,X_2,...\) with mean \(\mu\) . Then it holds:

\(lim_{n\rightarrow \infty}\frac{1}{n}\sum^{n}_{i=1}X_i\rightarrow \mu\)

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Central limit theorem 

Given a sequence of independent and identical distributed random variables \(X_1,...,X_n\) with expected value \(\mu\) and variance \(\sigma^2\) , then the distribution of \(S_n=\frac{1}{n}(X_1+...+X_n)\)is approximately normal with mean \(\mu\) and variance \(\frac{1}{n}\sigma^2\) or, 

\(\sqrt{n}\left(\frac{1}{n}\sum\limits^n_{i=1}X_i-\mu \right)\xrightarrow d\mathcal{N}(0,\sigma)\)

the \(\xrightarrow d\) reads "converges in distribution to".

How large does n to be chosen? Depend on the underlying distributions of the sample sequences. 

 

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Chebychev's inequality 

For any random variable and c>0 there holds:

\(P(|X-E(X)|\geq c)\leq\frac{Var(X)}{c^2}\)

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Empirical vs. theoretical quantities

Quantities estimated from a given sample are often referred to as empirical or sample quantities. \(\hat{\;\mu} \)

The corresponding true or model quantities are often referred to as the theoretical\(\mu\)

 

Given a sample \(x_1,...,x_N\)of a random variable X. Consider a parameter \(\Theta\) on X, e.g., the mean \(\mu\) .

Then the estimator \(\hat{\;\Theta}\) is a function of a sample (i.e., is a statistics) of the random variable X which assigns to the sample values which distribution depend on (and are close to) \(\Theta\) .