Probablity Density Functions
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Transcript of Probablity Density Functions
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Probabilty Density Function of the “Normal Distribution”
Source: Wikipedia: http://en.wikipedia.org/wiki/Normal_distribution
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Probabilty Density Function of the “Chi-Square Distribution”
Source: Wikipedia: http://en.wikipedia.org/wiki/Chi-squared-distribution
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Probabilty Density Function of the “Gamma Distribution”
Source: Wikipedia: http://en.wikipedia.org/wiki/Gamma-distribution
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x
x
f(x) =
a b
1/(b-a)
Area =1
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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.
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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
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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
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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)
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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
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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
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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