**Objective:**

The
main aim of this topic is to study and observe the difference between the
normal distribution and lognormal distribution using R commands. In financial
time series analysis it is more appropriate to use the lognormal distribution
than the normal distribution.

**Introduction:**

The
moments of a normal distribution and lognormal distribution are not exactly the
same and hence we observe a shift in log normal from a normal distribution.

The
above formulae and further characteristics are also available on wikipedia
using the following link - http://en.wikipedia.org/wiki/Log-normal_distribution.The
wiki article also elaborates on the PDF and CDF which are covered in brief in
this article. The mean of the distribution shifts by 0.5*variance, hence you
can observe that the graph would shift with shift in variance from .5 to 1.
This is clear from the image below.

**R functionality:**

The above
plot is created in R using the following commands. Also, in R the plot may get
cut while plotting. The best way to plot all the images correctly is plot the
image with the maximum y value first using the PLOT command and subsequent
images using the lines command in R.

x<- seq(0,3,length = 100)

plot(x, dlnorm(x,0, 0.25), type=
"l", col ="blue")

lines(x,dlnorm(x,0,0.5), type ="l",
col = "red")

lines(x,dlnorm(x,0,1),
type ="l", col = "green")

dlnorm(n, mean, standard deviation) = computes the density
of a lognormal distribution

type ="l" = allows r to plot a line graph

col = is used to color the line

lwd = 2 = can also be used in the plot to increase the width
of the plot

**Cumulative distribution function of a log normal in R:**

x<- seq(0,3,length =
100)

plot(x,
plnorm(x,0, 0.25), type ="l", col ="blue", lwd = 2)

lines(x,plnorm(x,0,1),
type ="l", col = "green")

lines(x,plnorm(x,0,0.5),
type ="l", col = "red")

In order to calculate the CDF of a log normal simply use
the command PLNORM in r.

PLNORM(n, mean, standard dev.) = computes CDF for a log
normal distribution.

RLNORM(n, mean , standard dev) = generates a data set with a given mean and Standard deviation.

qlnorm(p, mean, standard deviation) = can be used to generate p quantiles of the log normal distribution

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