Lernkarten

continuous random variables: 

Random variables X and Y are independent if for any x and y:

\(P(X\leq x, Y\leq y)=P(X\leq x)P(Y\leq y)=F(x)G(y)\)

where F(x) and G(x) are the corresponding CDFs.

discrete random variables:

Random variables X and Y are independent if for any \(x_i\)and \(y_i\):

\(P(X\leq x_i,Y\leq y_j)=P(X\leq x_i)P(Y\leq y_j)\)

Percentile: quantiles expressed in percentages: The 0.2 quantile is the 20th percentile

Quartiles: are 25th and 75th percentiles 

Median: is the 0.5-quantile

The nth moment \(\mu_n\) of a probability density \(f_X(x)\) is defined as:

• (cont.):  \(\mu_n=E(X^n)=\int x^n*f_X(x)dx\)
• (discr.):  \(\mu_n=E(X^n)=\sum x^n_k * f_X(x_k)\)

The n th central moment \(\mu'_n\) of a probability density \(f_X(x)\) is defined with respect to the first moment (\(\mu\)) as

 \(\mu_n'=E((X-\mu)^n)=\int (x-\mu)^n * f_X(x)dx \)

The expected value, also called the mean is defined as the first moment:

\(\mu=E(x)=\int x*f(x)dx \)

The expected value can be physically seen as the centroid of mass in physics.

The variance is defined as the second central moment:

\(\sigma^2=Var(x)=E((X-\mu)^2)=E(X^2)-\mu^2\)

The variance gives the spread around the expected value. 

The skewness is the third central moment divided by the standard diviation to the third power:

\(.\\\gamma_1=E[(\frac{X-\mu}{\sigma })^3 ]=\frac{E[(X-\mu )^3]}{(E[(X-\mu )^2])^{3/2}}\)

Kurtosis (measure of peakness)

The kurtosis of any univariate normal distribution is 3. It is common to compare the kurtosis of a distribution to this value. Distributions with kurtosis less than 3 are said to be platykurtic, although this does not imply the distribution is "flat-topped" as sometimes reported. Rather, it means the distribution produces fewer and less extreme outliers than does the normal distribution. An example of a platykurtic distribution is the uniform distribution, which does not produce outliers.

Der Exzess gibt die Differenz der Wölbung der betrachteten Funktion zur Wölbung der Dichtefunktion einer normalverteilten Zufallsgröße an.

The mode is the value that appears most often in a set of data. For a continuous probability distribution it is the peak. 

probability density function (PDF):

\(f(x)=\frac{1}{b-a}\;for\;x\;\epsilon\;[a,b]\)

cumulative distribution function (CDF):

\(F_X(x)=\left\{ \begin{array}{c} 0\;\;\;for\;x< a\\\frac{x-a}{b-a}\;\;\;for\;\;x\;\epsilon\;[a,b]\\1\;\;\;for\;x\geq b \end{array} \right.\)