The y-bounded r-th raw moment of the Lognormal distribution equals:
μyr=e2r(rVar2+2μ)[1−Φ(Varlny−(rVar2+μ))]
Examples
## The zeroth truncated moment is equivalent to the probability functionplnorm(2, meanlog =-0.5, sdlog =0.5)mlnorm(truncation =2)## The (truncated) first moment is equivalent to the mean of a (truncated) random sample,#for large enough samples.x <- rlnorm(1e5, meanlog =-0.5, sdlog =0.5)mean(x)mlnorm(r =1, lower.tail =FALSE)sum(x[x > quantile(x,0.1)])/ length(x)mlnorm(r =1, truncation = quantile(x,0.1), lower.tail =FALSE)