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Probabilty Density Function of the “Normal Distribution”

Source: Wikipedia: http://en.wikipedia.org/wiki/Normal_distribution

Probabilty Density Function of the “Chi-Square Distribution”

Source: Wikipedia: http://en.wikipedia.org/wiki/Chi-squared-distribution

Probabilty Density Function of the “Gamma Distribution”

Source: Wikipedia: http://en.wikipedia.org/wiki/Gamma-distribution

x

x

f(x) =

a b

1/(b-a)

Area =1

Averaging two independent random variables

{x1,1 , x1,2 , … x1,n } n: sample size (n=10000)

{x2,1 , x2,2 , … x2,n } Two independently drawnrandom number sets X1X2 from uniform distributions.

Averaging two independent random variables{x1,1 , x1,2 , … x1,n } n: sample size (n=10000)

{x2,1 , x2,2 , … x2,n }

Two independently drawnrandom number sets X1X2 from uniform distributions.

Averaged:

Y2,1=(X1,1+X2,1)/2

Averaging independent random variables{x1,1 , x1,2 , … x1,n } n: sample size (n=10000)

…{x5,1 , x5,2 , … x5,n }

Five independently drawnrandom number sets X1X5 from uniform distributions.

Averaged:

Y5,i=(X1,i+X2,i+ X3,i+X4,i + X5,i)/5

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 uniformlydistributed variablesaveraged (10000 times)

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 ofthe average is a function

of the sample size.

nsum decreasing

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

Scatter-Plots are simple but yetvery powerful presentations oftwo variables and how they are related.

Pairs of vectors can be plottedin R in this way. In our case the time (year and month) gives a natural order to the data. The vector elements atthe same position are formingthe coordinates for the x and y axis.

The vector with x-coordinates is y1,The y-coordiantes are in vector y2