Wednesday, 25 April 2012

Log Normal Distribution

if $X$ is Gaussian Distributed Random variable i.e., if
\begin{align*}
X \sim  \mathcal{N}(\mu,\sigma^2) \\
Y=e^X\\
\end{align*}
Y is said to be Log Normally Distributed i.e.,
\begin{align*}
Y \sim Log-\mathcal{N}(\mu,\sigma^2)
\end{align*}
If MGF of X is Given i.e., $\Phi_X(s)$ is Given.

Find the nth Moment of Y, where $n \in \mathbb{Z}$, Without Finding the PDF of Y

1. This is one of the Convenient Way to find the Moments of Log Normal Distribution, if MGF of Normal Distribution is in our hands. Its a Simple Trick.

2. Some facts about Log Normal Random variable:
\begin{align*}
if \; Y \sim Log-\mathcal{N}(\mu,\sigma^2)\\
then \; \frac{1}{Y} \sim Log-\mathcal{N}(-\mu,\sigma^2) \\

if\; Y_j \sim Log-\mathcal{N}(\mu_j,\sigma_j^2)\\
\forall j=1 \cdots N \; \\
and \; Y_i, Y_j \;\text{are Statistically Independent}\; \\
\forall (i \neq j)=1 \cdots N \\

Then\; Z= \left(\prod _{j=1} ^{N}Y_j \right) \sim Log-\mathcal{N}\left(\sum \mu_j,\sum \sigma_j^2 \right )
\end{align*}

3. So This Log Normal Distribution is One Example, whose Reciprocal is also Log Normally Distributed. There is one More Distribution whose Reciprocal has the same Distribution.It is Cauchy Distribution. Lets take Standard Cauchy Random variable $X$.

$$f_X(x)=\frac{1}{\pi\,(1+x^2)}\,\, -{\infty}<x<{\infty}$$

Then PDF of $$Y=\frac{1}{X}$$ is given by:

$$f_Y(y)=\frac{1}{\pi\,(1+y^2)}\,\, -{\infty}<y<{\infty}$$