Lernkarten

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

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) 

Yes, then:

• the distribution only consists of one constant 
• mean, median and mode are the same 

The Cauchy distribution 

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

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

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

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. 

For any random variable and c>0 there holds:

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

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) \(\Theta\) .