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Independent random variables 

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 \(y_i\):

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

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Define the expressions Quantile, Percentile, Median and Quartile 

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

 

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What is a moment? 

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

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

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How is the expected value and the variance defined?

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. 

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Explain Skewness! 

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

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

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What is the fourth central moment? 

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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.

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What is the Mode?

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

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What are the probability density and the cumulative distribution function of the uniform distribution?

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probability density function (PDF):

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

cumulative distribution function (CDF):

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