#
CSI 779 / STAT 789

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Characteristics of Probability Densities; Q-Q Plots

For most continuous random variables we can write a
*probability density function (PDF), *
which is yields the relative probability that the random variable takes
values in given intervals.
Three of the most important characteristics of random variables
are

the range
the moments (mean, variance, etc.) and
the shape of the density (assuming it exists).
The normal distribution is the standard for comparison. The shapes of
distributions may be

normal
skewed
heavy-tailed
light-tailed
bimodal (or multimodal)
or not
The relative frequencies and the order statistics of a
random sample from a given distribution will be "similar to" those of
the parent distribution.

A histogram and a q-q plot of the sample
are useful in assessing the shape.

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Normal: xx<-rnorm(500)

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Skewed; Lognormal: xx<-rlnorm(500)

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Heavy-Tailed; t(5): xx<-rt(500,5)

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Light-Tailed

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Bimodal:
xx[1:250]<-rnorm(250,-2)
xx[251:500]<-rnorm(250,2)

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Mixture that is not bimodal:
xx[1:250]<-rnorm(250,-1)
xx[251:500]<-rnorm(250,1)