# Suppose Y t given X t has an exponential density with mean −1 /X t ( X t < 0).

The exponential case

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Suppose Y t given X t has an exponential density with mean −1 /X t ( X t < 0).

Just from $13/Page

Suppose *Y _{t}*given

*X*has an exponential density with mean −1

_{t}*/X*(

_{t}*X*0). The observation density is given by

_{t}<*p(y _{t}*|

*x*= exp{

_{t})*y*+ ln

_{t}x_{t}*(*−

*x*}

_{t})*, y*0

_{t}>*,*

which has the form with *b(x) *= −ln*(*−*x) *and *c(y) *= 0. The state densities

corres ponding to the family of conjugate priors are given by

*p x _{t}*

_{+1}|y

*= exp{*

^{(t)}*α*

_{t}_{+1}|

*t*

^{x}_{t}_{+1}−

*λ*

_{t}_{+1|}

_{t}*b(x*+1

_{t}*)*+

*A*

_{t}_{+1|}

*}*

_{t}*,*−∞

*< x <*0

*.*