Relationship between dist. of x and dist. of 1/x?

  1. Mapes

    Mapes 2,532
    Science Advisor
    Homework Helper
    Gold Member

    I've been working recently in the area of tissue cell mechanics; specifically, I'm measuring mechanical stiffness (or compliance, the reciprocal of stiffness) and considering its possible underlying distribution.

    I was wondering about the following: If the distribution of stiffness measurements is approximately Gaussian (or lognormal, or gamma distributed, etc.), then what can we say about the distribution of the corresponding compliance (= 1/stiffness) values? More generally, if [itex]x[/itex] is distributed in a certain way, what about [itex]1/x[/itex]? Is there a simple relationship?
     
  2. jcsd
  3. statdad

    statdad 1,455
    Homework Helper

    No - there is no universal statement that can be made. Each case needs to be considered on its own.
    (The mathematical ideas behind studying the distributions is the same in each case, but unless I'm totally off that wasn't the point of your inquiry.)
     
  4. Redbelly98

    Redbelly98 12,029
    Staff Emeritus
    Science Advisor
    Homework Helper

    There is a relationship, how simple depends on the details of your example.

    Given a probability distribution f(x), we seek the distribution g(y) where y is a function of x. A simple probability conservation argument tells us that

    f(x) |dx| = g(y) |dy|​

    so that

    g(y) = f(x) / |dy/dx|​

    Take y = 1/x, and f(x) is whatever you think, you can get g(y).


    EDIT:
    Continuing the example for y = 1/x

    Since |dy/dx| = 1/x2 = y2, we have

    g(y) = f(x) / y2

    And, of course, you would substitute 1/y for x in the expression for f(x).
     
    Last edited: Nov 2, 2009
  5. statdad

    statdad 1,455
    Homework Helper

    The transformation approach is correct (modulo being careful around x = 0); my intention was to say there is nothing simple to say about the type of distribution for X and the type for 1/X (normal to normal, t to t, etc).
     
  6. Redbelly98

    Redbelly98 12,029
    Staff Emeritus
    Science Advisor
    Homework Helper

    True :smile:
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?