if X is Gaussian Distributed Random variable i.e., if
X∼N(μ,σ2)Y=eX
Y is said to be Log Normally Distributed i.e.,
Y∼Log−N(μ,σ2)
If MGF of X is Given i.e., ΦX(s) is Given.
Find the nth Moment of Y, where n∈Z, Without Finding the PDF of Y
X∼N(μ,σ2)Y=eX
Y is said to be Log Normally Distributed i.e.,
Y∼Log−N(μ,σ2)
If MGF of X is Given i.e., ΦX(s) is Given.
Find the nth Moment of Y, where n∈Z, Without Finding the PDF of Y
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.
ReplyDeleteSome facts about Log Normal Random variable:
ifY∼Log−N(μ,σ2)then1Y∼Log−N(−μ,σ2)ifYj∼Log−N(μj,σ2j)∀j=1⋯NandYi,Yjare Statistically Independent∀(i≠j)=1⋯NThenZ=(N∏j=1Yj)∼Log−N(∑μj,∑σ2j)
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.
fX(x)=1π(1+x2)−∞<x<∞
Then PDF of Y=1X
fY(y)=1π(1+y2)−∞<y<∞