# Simulation of Lognormal distribution

This post is a try about how yo simulate lognormal distribution in R. Lognormal distribution is used a lot in cumulative data (e.x. counting), which is very similar with normal distribution except x should be larger than 0. For instance, the number of schools, the number of students. But I always have no idea about how to parameterized this distribution. I’ll update this post as I learn more about lognormal distribution…

## Simulate lognomal distibution

### Simulation Study 1

This study is to simulate lognormal density distribution based on mean and sd of depedent variable (Y). My simulated mean of y is 891, and sd is 490, N (sample size) is 200000. Then use the formular below:

mu  = log(m^2/phi) # log mean
sigma = sqrt(log(1+v/m^2)) # log sd


It could calculate the log mean and log standard deviation for lognormal distribution.

set.seed(20171108)

#### Give Y mean and Y sd, simluate lognormal distribution data.
m = 891 # geometric mean
sd = 490 # geometric sd
v = sd ^ 2
phi = sqrt(v + m^2)

mu    = log(m^2/phi) # log mean
sigma = sqrt(log(1+v/m^2)) # log sd

y <- rlnorm(n = 200000, mu, sigma) %>% round(0)
m.sim <- mean(y) # should be close to 891
sd.sim <- sd(y) # should be close to 490

row1 <- c(m, mu,m.sim)
row2 <- c(sd, sigma,sd.sim)
table <- rbind(row1, row2)
colnames(table) <- c("Original", "Log", "Simulated")
rownames(table)  <- c("Mean", "SD")
kable(table)


This is the original, log and simulated mean and sd. It could be easily found that simulated ones are very closed to original.

Original Log Simulated
Mean 891 6.660225 891.1468
SD 490 0.514041 490.4396
plot(density(y))


From the density plot below, we can see the mean of X is also close to 891.