1. Distributions and Densities
Jacek Wallusch _________________________________ Mathematical Statistics for International Business
Lecture 5:
Distributions and Densities

2. is the distribution skewed or symmetrical?
getting started
Income Distribution in Germany:
what should we look at?
shape of distribution
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
is the distribution skewed or symmetrical?
Mathematical Statistics: 5
What is the intuitive definition of distribution?

3. frequencies and income clusters
Distribution ____________________________________________________________________________________________
example
Income Deciles in Germany:
frequencies and income clusters
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 5
black – 1934, red – 2006

4. Distribution Function ____________________________________________________________________________________________
Important definition
Random variable:
continuous
e.g.:
discrete
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 5 4

5. Distribution Functions:
CDF&PDF
Distribution Functions:
cumulative (CDF):
probability (PDF):
probability that random variable X will be not greater than x
probability that random variable X will fall within the interval between x and x + dx
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 5 5

6. finite x, continuous or discrete random variable X
Distribution Function ____________________________________________________________________________________________
CDF
Cumulative Distribution Function:
finite x, continuous or discrete random variable X
Properties of CDF:
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 5
Recall: probability measure, semi-closed interval, improper integrals;

7. Distribution and Density ____________________________________________________________________________________________
properties
Properties of a distribution of discrete R.V.:
Properties of a density function:
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 5
Exercise 1: Why b.-properties must be satisfied?
Exercise 2: eq(4.5) p. 46

8. Empirical Distribution ____________________________________________________________________________________________
Histogram
Histogram:
graphical presentation of frequencies or relative frequencies for selected clusters (bins) of a random variable
Example:
Discount granted:
frequency
relative frequency
less than 20% 5
0.040
20%-25% 22
0.176
25%-30% 45
0.360
30%-35% 35
0.280
35%-40% 15
0.120
more than 40% 3
0.024
Total: 125
Total: 1.000
Probability distribution and uncertainty and risk – this topic will be reconsidered soon
Mathematical Statistics: 5 8