1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Statistics, inverse of cdf

  1. Jul 22, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that the given function is a cdf (cumulative distribution function) and find [itex]F_X^{-1}(y)[/itex]
    (c) [itex]F_X(x) = \frac {e^{x}}4 [/itex], if [itex]x<0[/itex], and [itex]1-(\frac {e^{-x}}4) [/itex], if [itex]x \geq 0 [/itex]


    2. Relevant equations

    for a strictly increasing cdf, [itex] F_X^{-1}(y) = x \iff F_X(x) = y [/itex]

    and for a non-decreasing (a.k.a. difficult problem) cdf, [itex] F_X^{-1}(y) = inf \{ x: F_X(x) \geq y \} [/itex]

    3. The attempt at a solution
    It's not so hard to show that F is a cdf. The [itex] lim_{x \to -\infty} F_X(x)= 0[/itex], the [itex]lim_{x \to \infty} F_X(x) = 1 [/itex], the function is non-decreasing, and right-continuous.

    I have the solution for the inverse, but it doesn't seem right to me. The given solution is

    [itex]F_X^{-1}(y) = ln(4y) [/itex] for [itex]0<y< \frac 14[/itex] and [itex] -ln(4(1-y))[/itex] for [itex] \frac 14 \leq y<1 [/itex]

    But this solution doesn't seem to agree with the definition of inverse F or the inverses I found.

    so if [itex] y = e^{\frac {x}4} [/itex], then doesn't this imply [itex] x = 4lny [/itex]? and doesn't [itex] y= 1 - (e^{\frac {-x}4 })[/itex], imply that [itex] x = -4ln(1-y) [/itex]?
    For example, using the inverses I have, the set of x's such that [itex]F_X(x) \geq \frac 14 [/itex] would include [itex] 4ln( \frac 12) \approx -2.772 [/itex] and [itex] -4ln(1- \frac 12) \approx 2.772 [/itex], and the infimum of the set (greatest number that is less than all other numbers in the set, I think) is -2.772. So I should use the first inverse, [itex] F_X^{-1}(y) = 4lny [/itex], since this function would have the smallest x's.

    Am I right here? Please help.
     
    Last edited: Jul 22, 2014
  2. jcsd
  3. Jul 22, 2014 #2
    I see my mistake now.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Statistics, inverse of cdf
  1. Meaning of Inverse CDF (Replies: 0)

  2. Statistics (CDF) (Replies: 6)

Loading...