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!

Homework Help: Conditional exponential distribution and exponential evidence

  1. Apr 3, 2013 #1
    1. The problem statement, all variables and given/known data
    This is a subset of a larger problem I'm working on, but once I get over this hang up I should be good to go. I have a set of measurements [itex]x_n[/itex] that are exponentially distributed

    [tex]p(x_n|t)=e^{-(x_n-t)} I_{[x_n \ge t]}[/tex]

    and I know that t is exponentially distributed as


    2. Relevant equations
    marginal probability
    [tex]p(x)=\int p(x|t) p(t) dt[/tex]

    3. The attempt at a solution
    So the probability of N observations of x are
    [tex]p(\mathbf{x}|t)=e^{-s(x)} e^{Nt} I_{[\textrm{min}(x_n) \ge t]}[/tex]
    [tex]s(x)=\sum_{n=1}^N x_n[/tex]

    Which means that
    [tex]p(\mathbf{x},t)=e^{-s(x)} e^{t(N-1)} I_{[\textrm{min}(x_n) \ge t]} I_{[t\ge0]}[/tex]

    If I want to find p(x) it should be
    [tex]p(\mathbf{x})=\int_0^{x_{min}} e^{-s(x)}e^{t(N-1)} I_{[\textrm{min}(x_n) \ge t]}I_{[t\ge0]} dt[/tex]
    [tex]p(\mathbf{x})=e^{-s(x)}\frac{1}{N-1}e^{t(N-1)}|^{t=x_{min}}_{t=0}I_{[\textrm{min}(x_n) \ge t]}I_{[t\ge0]}[/tex]
    [tex]p(\mathbf{x})=e^{-s(x)}\frac{1}{N-1}I_{[\textrm{min}(x_n) \ge t]}I_{[t\ge0]}(e^{x_{min}(N-1)}-1)[/tex]

    The issue is that this function isn't normalized. Are my limits wrong, or should I renormalize?
    Last edited: Apr 3, 2013
  2. jcsd
  3. Apr 4, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    The formula for ##p(\mathbf{x})## should not have ##t## in it.

    Anyway, why would you need to re-normalize? Your ##p(\mathbf{x})## integrates to 1 when integrated over ##\mathbb{R}_{+}^N##. If you don't believe it, try the simple cases of N = 2 and N = 3 first.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted