Distributions and Densities: Gamma-Family and Beta Distributions Jacek Wallusch _________________________________ Mathematical Statistics for International Business Lecture 5: Distributions and Densities: Gamma-Family and Beta Distributions

Standard deviation, describes the spread around the mean Normal Distribution ____________________________________________________________________________________________ Properties Normal PDF: Parameters: mu (m): sigma (s): Mean value (average), describes the location of the distribution’s midpoint Standard deviation, describes the spread around the mean Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 Data: Excel file forex.xls

Normal Distribution ____________________________________________________________________________________________ Properties Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 File: http://wallusch-datenbank.de/resources/quants.rar

mean values and variances Normal Distribution ____________________________________________________________________________________________ central moments Location and dispersion again: mean values and variances Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 black – X~N(0,0.25), green – X~N(0,0.1406), red – X~N(0,0.0625) MS Excel function: NORM.DIST

positive skewness and nonnegative values of the random variables Gamma Distribution ____________________________________________________________________________________________ Gamma function Properties: positive skewness and nonnegative values of the random variables Formula: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 Data: Excel file forex.xls

Gamma Distribution ____________________________________________________________________________________________ Gamma function Parameters: alpha: beta: Shape parameter affecting the shape of the distribution Rate parameter affecting the spread of the distribution theta: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Scale parameter affecting the spread of the distribution large value = = small spread large value = = large spread Mathematical Statistics: 5 Data: Excel file forex.xls

Gamma Distribution ____________________________________________________________________________________________ example Organizational culture: variable Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 Data Courtesy: Monika Olszewska

Gamma Distribution ____________________________________________________________________________________________ c2 distribution Gamma distribution: Characteristics of c2 distribution: degrees of freedom – minimum number of independent random variables describing the system and influencing the result Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 k – degrees of freedom

Gamma Distribution ____________________________________________________________________________________________ c2 distribution ...or in other words: define the sequence X define W then: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 k – degrees of freedom

c2 Distribution ____________________________________________________________________________________________ generator Excel function: =CHIDIST(X;df) =ROZKŁAD.CHI(X;df) =ХИ2РАСП(x;степени_свободы) X – value, at which the distribution should be evaluated Probability distribution and uncertainty and risk – this topic will be reconsidered soon df – number of degrees of freeedom Mathematical Statistics: 5 k – degrees of freedom

Gamma Distribution ____________________________________________________________________________________________ c2 distribution distribution Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 d.f. – degrees of freedom

Gamma Distribution ____________________________________________________________________________________________ c2 distribution distribution Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 d.f. – degrees of freedom

Gamma Distribution ____________________________________________________________________________________________ F distribution define first: define a new variable: then: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 k – degrees of freedom

Gamma Distribution ____________________________________________________________________________________________ Student t distribution define first: define a new variable: then: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 k – degrees of freedom

Defined over the closed interval between 0 and 1 Beta Distribution ____________________________________________________________________________________________ Beta function Properties: Defined over the closed interval between 0 and 1 Formula: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 Suitable to model proportions

Beta Distribution ____________________________________________________________________________________________ Example Example: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5

Poisson Distribution ____________________________________________________________________________________________ Properties Properties: Defined for integers Formula: Probability distribution and uncertainty and risk – this topic will be reconsidered soon Mathematical Statistics: 5 Rare events