1. Probablity Density Functions

2. Gaussian Probability Density function (NORMAL Distribution)
Probabilty Density Function of the “Normal Distribution”
Source: Wikipedia:

3. Other Important PDFs Probabilty Density Function of the
“Chi-Square Distribution”
Source: Wikipedia:

4. Probabilty Density Function of the
Other Important PDFs
Probabilty Density Function of the
“Gamma Distribution”
Source: Wikipedia:

5. UniForm Distribution x
f(x) =
1/(b-a)
Area =1 a b x

6. Averaging Averaging two independent random variables
{x1,1 , x1,2 , … x1,n } n: sample size (n=10000)
{x2,1 , x2,2 , … x2,n }
Two independently drawn
random number sets X1
X2 from uniform distributions.

7. Averaging random variables
Averaging two independent random variables
{x1,1 , x1,2 , … x1,n } n: sample size (n=10000)
{x2,1 , x2,2 , … x2,n }
Two independently drawn
random number sets X1
X2 from uniform distributions.
Averaged:
Y2,1=(X1,1+X2,1)/2

8. Averaging Random variables
Averaging independent random variables
{x1,1 , x1,2 , … x1,n } n: sample size (n=10000) …
{x5,1 , x5,2 , … x5,n }
Five independently drawn
random number sets X1
X5 from uniform distributions.
Averaged:
Y5,i=(X1,i+X2,i+ X3,i+X4,i + X5,i)/5

9. Averaging random VARIABLEs
The shape and width of distribution in the histogram of the averages changes
with the number of variables entering the averaging calculation.
Note that the average itself is a random variable and has a mean and standard deviation.
30 uniformly
distributed
variables
averaged
(10000 times)

10. Averaging random VARIABLEs
The shape and width of the distribution in the histogram of the averages changes
with the number of variables entering the averaging calculation.
Note that the average itself is a random variable and has a mean and standard deviation.
Standard deviation σ of
the average is a function
of the sample size.
nsum decreasing

11. Summary:
During the averaging of randomly sampled data the distribution shape converges towards a Gaussian Distribution with increasing sample size.
The larger the sample the smaller the standard deviation σ (i.e. the smaller the uncertainty of the average value).
When the samples that enter the averaging are independent, then

12. Analysis of reLationships between two variables
Scatter-Plots are simple but yet
very powerful presentations of
two variables and how they are
related.
Pairs of vectors can be plotted
in R in this way. In our case the time (year and month) gives a natural order to the data. The vector elements at
the same position are forming
the coordinates for the x and y axis.
The vector with x-coordinates is y1,
The y-coordiantes are in vector y2