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Albert
Posts:
4
Registered:
5/6/11


Question: Normal Distribution Functions and Product
Posted:
May 9, 2011 10:30 PM


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 Lognormal 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.



