rr.mar <- read.csv("http://www.stat.ufl.edu/~winner/data/rocknroll_marathon_mf2015a.csv", header=T) attach(rr.mar); names(rr.mar) mean(mph); sd(mph) tapply(mph,Gender,mean) tapply(mph,Gender,sd) hist(mph,breaks=50) beta <- mean(mph)/var(mph) alpha <- beta*mean(mph) ast <- log(alpha) bst <- log(beta) n <- length(mph) xmph <- seq(4,11,.01) ymph <- dgamma(xmph,alpha,beta) hist(mph,ylim=c(0,180),breaks=seq(4,11,1/6)) lines(xmph,ymph*length(mph)/6) m.mph <- mph[Gender=="M"] f.mph <- mph[Gender=="F"] m.beta <- mean(m.mph)/var(m.mph) m.alpha <- m.beta*mean(m.mph) (m.s2 <- log((var(m.mph)+mean(m.mph)^2)/mean(m.mph)^2)) (m.mu <- log(mean(m.mph)) - 0.5*m.s2) m.xmph <- seq(4,11,.01) mg.ymph <- dgamma(m.xmph,m.alpha,m.beta) #ml.ymph <- dlnorm(m.xmph,mean(log(m.mph)),sd(log(m.mph))) ml.ymph <- dlnorm(m.xmph,m.mu,sqrt(m.s2)) f.beta <- mean(f.mph)/var(f.mph) f.alpha <- f.beta*mean(f.mph) (f.s2 <- log((var(f.mph)+mean(f.mph)^2)/mean(f.mph)^2)) (f.mu <- log(mean(f.mph)) - 0.5*f.s2) f.xmph <- seq(4,11,.01) fg.ymph <- dgamma(f.xmph,f.alpha,f.beta) #fl.ymph <- dlnorm(f.xmph,mean(log(f.mph)),sd(log(f.mph))) fl.ymph <- dlnorm(f.xmph,f.mu,sqrt(f.s2)) par(mfrow=c(2,2)) hist(m.mph,ylim=c(0,120),breaks=seq(4,11,1/6),main="Males/Gamma") lines(m.xmph,mg.ymph*length(m.mph)/6) hist(f.mph,ylim=c(0,120),breaks=seq(4,11,1/6),main="Females/Gamma") lines(f.xmph,fg.ymph*length(f.mph)/6) hist(m.mph,ylim=c(0,120),breaks=seq(4,11,1/6),main="Males/Lognormal") lines(m.xmph,ml.ymph*length(m.mph)/6) hist(f.mph,ylim=c(0,120),breaks=seq(4,11,1/6),main="Females/Lognormal") lines(f.xmph,fl.ymph*length(f.mph)/6) #hist(log(f.mph),ylim=c(0,120),breaks=seq(1.4,2.2,1/50))