1. Jacek Wallusch _________________________________ Statistics for International Business Lecture 8: Distributions and Densities

2. Gamma Distribution ____________________________________________________________________________________________ Statistics: 8 Gamma function Data: Excel file forex.xls Properties: positive skewness and nonnegative values of the random variables Formula:

3. Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution k – degrees of freedom Gamma distribution: Characteristics of : Characteristics of  2 distribution : degrees of freedom – minimum number of independent random variables describing the system and influencing the result

4. Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution k – degrees of freedom...or in other words: then: define the sequence X define W

5.  2 Distribution ____________________________________________________________________________________________ Statistics: 8 generator k – degrees of freedom Excel function: =ROZKŁAD.CHI(X;df) X – value, at which the distribution should be evaluated df – number of degrees of freeedom

6. Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution d.f. – degrees of freedom distribution

7. Gamma Distribution ____________________________________________________________________________________________ Statistics: 8  2 distribution d.f. – degrees of freedom distribution

8. Gamma Distribution ____________________________________________________________________________________________ Statistics: 8 F distribution k – degrees of freedom define first: define a new variable: then:

9. Gamma Distribution ____________________________________________________________________________________________ Statistics: 8 Student t distribution k – degrees of freedom define first: define a new variable: then:

10. Beta Distribution ____________________________________________________________________________________________ Statistics: 8 Beta  – left parameters,  – right parameters Distribution: Examples:

11. Calculating Probabilities ____________________________________________________________________________________________ Statistics: 8 Excel k – degrees of freedom, T - tails Excel Functions: calculating probability for known value of random variable X =ROZKŁAD.type(X;k;T) calculating the value of random variable X for known probability value =ROZKŁAD.type.ODW(Prob; k)

12. Calculating Probabilities ____________________________________________________________________________________________ Statistics: 8 distribution k – degrees of freedom t-Stat: =ROZKŁAD.T(X;k;Tails) =ROZKŁAD.T.ODW(Prob;k) F-Stat: =ROZKŁAD.F(X;k 1 ;k 2 ) =ROZKŁAD.F.ODW(Prob; k 1 ;k 2 )  2 -Stat: =ROZKŁAD.CHI(X;k) =ROZKŁAD.CHI.ODW(Prob; k)