I am trying to plot the beta-gumbel distribution using R(software) by the following, The genreal idea is that, in the pdf of beta distribution, instead of plugging in x, we use the cdf of gumbel instead. But I couldn't get the right plot.
x <- seq(-3, 3, length=100)
Fx = pgumbel(x,loc=0,scale=1)
y = dbeta(Fx,shape1=0.5,shape2=0.5)
plot(x, y, type="l", lty=2, xlab="x value", ylab="Density",ylim=c(0,1))
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