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 May 27th, 2008, 05:46 AM #1 Newbie   Joined: May 2008 Posts: 4 Thanks: 0 Is this in anyway solvable? Hi, For my Bsc Thesis, i'm trying to create random variables for a custom pdf (probability denisty function). The final step involves solving the cdf function: y = 1-(exp(-a*x)*(1+a*x) So what i'm trying to create is a function x = f(y) from y = f(x) However is this possible? I have been trying to crack it all day, maybe I'm just not seeing it. Thanks a million! May 27th, 2008, 06:07 AM #2 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms That looks like it could probably be solved with Lambert's W function. Would it help you to have a solution in terms of a special function? I don't suspect there's a closed-form solution in +, -, *, /, ^, log, and exp. May 27th, 2008, 10:11 AM   #3
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 Originally Posted by CRGreathouse That looks like it could probably be solved with Lambert's W function. Would it help you to have a solution in terms of a special function? I don't suspect there's a closed-form solution in +, -, *, /, ^, log, and exp.
I am not here for a quick and dirty solution I really want to learn how to solve these (if only because I will have to solve more of them before I can even think of finishing my thesis!).

I was frustrated and therefore used mathematica to solve it and indeed fould a solution that involved the productlog. However once I tested it the mathematica generated function did not return correct values!

It would help me to have a solution, so if you are able to solve I would greatly appreciate this! would it be possible to do in steps so I can follow and learn?

Ps. Should I have know how to do this? I have only had introductionary calculas.

Right now I'm going to read up on the Lambert's W/ Product log.

Thanks a million..  May 27th, 2008, 11:03 AM #4 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms I'll look into some material on the function myself to see if I can help. I've used it to solve problems, but I'll admit its use does not come fully naturally to me yet. As an aside: How can it be that you're writing your thesis, but have only taken introductory calculus? My college required introductory/advanced/multivariable calculus and real analysis (total 15 semester hours) for any math major, and even that seems flimsy to me since it avoids complex analysis.* * As I was not required, I never did take complex analysis. I still feel the lack. May 27th, 2008, 11:20 AM   #5
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 Originally Posted by CRGreathouse I'll look into some material on the function myself to see if I can help. I've used it to solve problems, but I'll admit its use does not come fully naturally to me yet. As an aside: How can it be that you're writing your thesis, but have only taken introductory calculus? My college required introductory/advanced/multivariable calculus and real analysis (total 15 semester hours) for any math major, and even that seems flimsy to me since it avoids complex analysis.* * As I was not required, I never did take complex analysis. I still feel the lack.
I guess it probably is because I study biology and they never thought it necessary. I have never seen a course on more advanced levels of math for us (I study in the Netherlands though not US).

I really wish I had the opportunity... cause I would of jumped on it! May 27th, 2008, 12:32 PM   #6
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Re: Is this in anyway solvable?

Quote:
 Originally Posted by Writhen Hi, For my Bsc Thesis, i'm trying to create random variables for a custom pdf (probability denisty function). The final step involves solving the cdf function: y = 1-(exp(-a*x)*(1+a*x) So what i'm trying to create is a function x = f(y) from y = f(x) However is this possible? I have been trying to crack it all day, maybe I'm just not seeing it. Thanks a million!
Is this some sort of population growth model? To my finance eyes it looks simply like a discounted cash flow but with a mismatch of continuous and discrete growth rates

i.e. let a= discount rate and x = time

then forgetting about the 1- term it is
(1+ax) / exp(ax)

so if a = 5% and t=2

it is 1.1 / 1.105 May 27th, 2008, 12:50 PM   #7
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 Originally Posted by Writhen I guess it probably is because I study biology and they never thought it necessary. Of course, because you're in biology not math. Sorry.

I'll test the function a but when I get home. May 28th, 2008, 02:48 AM   #8
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Re: Is this in anyway solvable?

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 Originally Posted by STV Is this some sort of population growth model? To my finance eyes it looks simply like a discounted cash flow but with a mismatch of continuous and discrete growth rates i.e. let a= discount rate and x = time then forgetting about the 1- term it is (1+ax) / exp(ax) so if a = 5% and t=2 it is 1.1 / 1.105
I can understand why you would think it is a population model but it’s not that. I will give a short history if you are interested.

I am modeling (pollutant) dispersal from multiple sources but to start only from a single point source.

I will use different probability density functions to model this. Though I thought I might start with the exponential: http://mathworld.wolfram.com/Exponentia ... ution.html

Cause it looked simple enough.

I suppose that this demonstrates dispersal in one single direction; however I am interested in dispersal along all directions (for now I’m assuming isotropy). So I arc-wise integrated it and renormalized it (= ensuring that the area under the curve is 1).

That resulted in this curve:

(a/(2*pi))*2*pi*x*a*exp(-a*x)

A second step would be to calculate its cumulative density function (cdf), I got this one:
y = 1-(exp(-a*x)*(1+a*x)

The last and final step would be to solve the x=cdf(y) function so I can use a random number generator to start stochastic “simulation�? of pollutant dispersal. And that is where I got stuck!

So that’s the history and today’s a new day.. so I’ll work on it again.

Thanks for the clues so far! Tags solvable Search tags for this page

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