In bgctw/lognorm: Functions for the Lognormal Distribution

Formula for Variance given sigma at log scale and expected value

We start with formulas for variance and mean. $$ V = (e^{\sigma^2} - 1) e^{2\mu + \sigma^2} \ m = e^{\mu + \sigma^2/2} $$ mean formula resolved for $\mu$ and substituted into V. $$ \mu = log_m - \sigma^2/2 \ V = (e^{\sigma^2} - 1) e^{2(log_m - \sigma^2/2) + \sigma^2} \ V = (e^{\sigma^2} - 1) e^{2log_m - \sigma^2 + \sigma^2} \ V = (e^{\sigma^2} - 1) e^{2log_m} \ V = (e^{\sigma^2} - 1) (e^{log_m})^2 \ V = (e^{\sigma^2} - 1) m^2 \ $$

n = 1e4
sigma = log(1.2)
#sigma = log(1.41)
mu = log(10)
logR = rnorm(n, mu, sigma)
R = exp(logR)
meanR = mean(R)
sdR = sd(R)
V2 = (exp(sigma^2) - 1)*exp(2*mu + sigma^2)
#
m = exp(mu + sigma^2/2)
V = (exp(sigma^2) - 1)*m^2
c(meanR, m)
c(sdR, sqrt(V), sqrt(V2))

xPred <- seq(-2,22,length.out = 101) 
plot(density(R), xlim = c(-2,22), lty = "dotted")
abline(v = meanR)
lines(dnorm(xPred, m, sqrt(V))~xPred, col = "blue", lty = "dashed")
lines(dlnorm(xPred, mu, sigma)~xPred, col = "green")

df <- data.frame(cv = c(0.05,0.1,0.2,0.5,1,2,5,10,20))
df$sigma = sqrt(log(df$cv^2 + 1))
df$sigmaStar <- exp(df$sigma)
#df$cvRev <- sqrt(exp(log(df$sigmaStar)^2) - 1)
df

plot(sigmaStar ~ cv, df[1:3,])

sigmaStar <- 1.2
(cv <- sqrt(exp(log(sigmaStar)^2) - 1))

bgctw/lognorm documentation built on March 17, 2021, 3:21 a.m.