CSI 779 / STAT 789

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.

    Normal: xx<-rnorm(500)


    Skewed; Lognormal: xx<-rlnorm(500)


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




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


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