My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
May 27th, 2008, 06:46 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!
Writhen is offline  
 
May 27th, 2008, 07:07 AM   #2
Global Moderator
 
CRGreathouse's Avatar
 
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.
CRGreathouse is offline  
May 27th, 2008, 11:11 AM   #3
Newbie
 
Joined: May 2008

Posts: 4
Thanks: 0

Quote:
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..



Writhen is offline  
May 27th, 2008, 12:03 PM   #4
Global Moderator
 
CRGreathouse's Avatar
 
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.
CRGreathouse is offline  
May 27th, 2008, 12:20 PM   #5
Newbie
 
Joined: May 2008

Posts: 4
Thanks: 0

Quote:
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!
Writhen is offline  
May 27th, 2008, 01:32 PM   #6
STV
Member
 
Joined: Aug 2007

Posts: 93
Thanks: 0

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
STV is offline  
May 27th, 2008, 01:50 PM   #7
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
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.
CRGreathouse is offline  
May 28th, 2008, 03:48 AM   #8
Newbie
 
Joined: May 2008

Posts: 4
Thanks: 0

Re: Is this in anyway solvable?

Quote:
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!
Writhen is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
solvable



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Is this question solvable? davedave Calculus 6 March 23rd, 2014 06:12 AM
Solvable Quintics mathbalarka Algebra 101 September 6th, 2012 11:03 PM
is it solvable ? versmart Algebra 8 September 15th, 2011 08:07 AM
Is this a solvable problem? gpxfiles77 Algebra 1 January 11th, 2010 04:44 PM
is it solvable ? versmart Abstract Algebra 5 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.