If ln(x) has a normal distribution N(0,1) how do I find and prove the distribution for x?

And if S(t)=S(0)exp((mu-(sigma^2)/2)t+sigma*sqrt(tao)*G), where G N(0,1) what is the density function for S(t)?

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- Apr 29th 2009, 05:57 AMmemanuelsonD
If ln(x) has a normal distribution N(0,1) how do I find and prove the distribution for x?

And if S(t)=S(0)exp((mu-(sigma^2)/2)t+sigma*sqrt(tao)*G), where G N(0,1) what is the density function for S(t)? - Apr 30th 2009, 12:02 AMmemanuelson
How do you guys do to make those nice equations?

- Apr 30th 2009, 09:40 AMMoo
Hello,

Click on the "picture" to see the code ;)

Anyway, there's a latex sub-forum, where you may be able to learn some things about writing equations.

But what is "tao" ?

Quote:

If ln(x) has a normal distribution N(0,1) how do I find and prove the distribution for x?

Now, I can never remember the method you guys use... But mine is similar, using the Jacobian of the transformation.

For any bounded and continuous function f, we have :

Make the substitution :

So

Thus the pdf of is , for x>0

Which is the pdf of the log-normal distribution, with parameters (0,1). - Apr 30th 2009, 11:59 PMmemanuelson
Thank you so much!