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Topic: Question: Normal Distribution Functions and Product
Replies: 1   Last Post: May 31, 2011 6:30 AM

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Albert

Posts: 4
Registered: 5/6/11
Question: Normal Distribution Functions and Product
Posted: May 9, 2011 10:30 PM
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Hello everyone,

Let me ask a couple of questions on the distribution functions.
Let me assume that E is normally distributed.
Its probability distribution function is as follows:
P(E)=1/(Sqrt[2 * Pi * SigmaE^2]) Exp[-( E - Em / Sqrt[2] *
SigmaE)^2]
where SigmaE^2 is variance, Em is Mean.
This is the fundamental expression for the Normal distribution, you
may already know.
Let me assume one more that T = B*Exp[E/A], where A and B are
constant.
Then, T is distributed following the Log-normal distribution
function.
I tried to derive it and the final expression that I obtained is as
follows:
P(T) = 1/(T * Sqrt[2 * Pi * SigmaT^2]) Exp[-( Ln [T - Tm] / Sqrt[2] *
SigmaT)^2]
where SigmaT = SigmaE/A, Tm = B*Exp[Em/A].
And, its range is from 0 to Infinity when it is integrated over T, I
mean, dT.
Now, let me slightly change the gear to ask a question that I am
wondering how to interpret.
1. Let me assume that E is normally distributed as I have mentioned
above.
Then, what is the distribution function for B, where B = C/E ?
C is a constant.
When I tried to solve it, the final expression for a distribution
function of B that I obtained is:
P(B) =  - 1/(B^2 * Sqrt[2 * Pi * SigmaE^2]) [C* Exp[-({1/B - 1/Bm} *
C / Sqrt[2] * SigmaE)^2]]
where Bm = C/Em.
Am I right? I am not sure since it showed negative probability due to
(-) sign, and I am wondering what it means (does it have a meaning or
I have to ignore it ?).
In addition, can anybody let me know how to determine the range of
the
P(B)?
I mean, range of the distribution function. (For the normal
distribution function, its range is from {-Infinity} to {+Infinity},
and I wrote the range for the lognormal distribution above, {0} to
{Infinity})
2. This is a more complicated question.
How can I derive the expression for the distribution function for T*B
assuming that E is normally distributed.
When T = B*Exp[E/A], P(T) follows lognormal distribution function,
and
when B = C/E, B may follow the above expression if I was correct.
Then, how about the distribution for Z (P(Z)), where Z = T*B ?
Albert.



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