Beta distributions
Contents
Beta distributions
For details on the beta distribution see WikiPedia:Beta distribution. Notice that thee distribution is symmetric for the parameters p, and q (α & β in the Wikipedia article formula), and that the x axis range is from 0 to 1, a probability.
Beta distributions have many shapes depending on the parameters. Their flexibility makes them useful for modelling uniform distributions &Beta(1,1) right through to symmetric or skewed distributions, or even U shaped distributions.
R programming language
For help on functions which create/manipulate beta distributions see;
?beta # or help(beta) ?rbeta # or help(rbeta) |
Usage examples are provided at;
example(beta) example(rbeta) example(dbeta) example(pbeta) example(qbeta) |
R code example 1
Lets generate a beta distribution using the rbeta built in R function.
quartz() # Alter the parameters n <- 1000 # Number of observations p <- 0.1 q <- 0.1 x <- rbeta(n, shape1= p, shape2=q)
breaks <- seq(0,1, length=21)
hist(x, breaks=breaks, freq=FALSE, main=paste("p=",p, ", q=", q, sep="")) lines(density(x, adjust=0.4), col="red")
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Notice α and β are less than one, so the plot is U shaped, as described in Wikipedia:Beta_distribution#Shapes.
Symmetry of α Β
R code example 2
quartz()
# Alter the parameters
n <- 10000 # Number of observations
for(i in 1:50) {
p <- q <- i
x <- rbeta(n, shape1= p, shape2=q)
breaks <- seq(0,1, length=21)
hist(x, breaks=breaks, freq=FALSE, main=paste("p=",p, ", q=", q, sep=""))
lines(density(x, adjust=0.4), col="red")
Sys.sleep(1)
}
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