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Mindscrape
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Homework Statement
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
[tex]p(t)=e^{-t}I_{[t\ge0]}[/tex]
Homework Equations
marginal probability
[tex]p(x)=\int p(x|t) p(t) dt[/tex]
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]
where
[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?
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