Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

CDF and MGF Relation

  1. Oct 29, 2009 #1

    Suppose that the Cumulative Distribution Function (CDF) of a random variable X is [tex]F_X(x)[/tex], which is by definition is:

    [tex]F_X(x)=\text{Pr}\left[X\leq x\right]=\text{Pr}\left[\frac{1}{X}\geq \frac{1}{x}\right]=1-\text{Pr}\left[\frac{1}{X}\leq \frac{1}{x}\right]=1-F_{1/X}\left(1/x\right)[/tex]

    Considering this relation between the CDF of X and the CDF of its reciprocal, what is the relation between the Moment Generating Function (MGF) of X and its reciprocal?

    Any help will be highly appreciated.

    Thanks in advance
  2. jcsd
  3. Oct 31, 2009 #2
    A good starting point would be to think of a relation between the CDF of X and the MGF of X, wouldn't it?
  4. Oct 31, 2009 #3
    Yes right, and I know what is the relation between them, but I want to see if another one has another idea. Anyway, the relation is:


    I have tried this, and it yields no where.

  5. Oct 31, 2009 #4
    What would the CDF and MGF look like if X is uniform on [0,1] ?
  6. Nov 27, 2009 #5
    The CDF of a uniformly distributed random variable X is:

    [tex]F_X(x)=\begin{cases}0&x<0\\\frac{x-a}{b-a}&a\leq x<b\\1&x\ge b\end{cases}[/tex]

    Here, it may easier to derive the MGF from the PDF, not from the CDF. The PDF of X will be:

    [tex]f_X(x)=\begin{cases}\frac{1}{b-a}&a\leq x\leq b\\0&\mbox{elsewhere}\end{cases}[/tex]

    Then the MGF of X is:


    But, what is the relation of this to the primary question?

    Anyway, I have found the following relations between the MGF of X and the MGF of its reciprocal:

    [tex]\mathcal{M}_{1/X}(s)=1-\sqrt{s}\int_0^{\pi/2}\frac{\sec^2 (\zeta)}{\sqrt{\tan (\zeta)}}J_1\left(2\,\sqrt{s\tan (\zeta)}\right)\mathcal{M}_X\left(\tan (\zeta)\right)\,d\zeta

    where [tex]J_v(.)[/tex] is the vth order bessel function of the first kind. I do not know how they got there. Does anybody know how to derive these relations?

    Thanks in advance
  7. Nov 27, 2009 #6


    User Avatar
    Science Advisor

    If X < 0 and x > 0, your statement about reciprocals doesn't hold.
  8. Nov 28, 2009 #7
    Yes , I forgot to mention that [tex]0\leq X, x<\infty[/tex]. Then, is there any problem?

  9. Nov 28, 2009 #8


    User Avatar
    Science Advisor

    Not in your original statement.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: CDF and MGF Relation
  1. Inequality of mgf (Replies: 2)