1. 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

2. 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

3. Normal Distribution ____________________________________________________________________________________________
Properties
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 5
File:

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

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

17. 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